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<title>Localization (commutative algebra)</title>

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Localization (commutative algebra)

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In <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a> and <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, localization is a formal way to introduce the "denominators" to a given <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> or <a href="Module_(mathematics)" title="Module (mathematics)">module</a>. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of <a href="Algebraic_fraction" title="Algebraic fraction">fractions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m}{s}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {m}{s}},}</annotation>

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</math><img src="./b50cf47f75892dda1b00c35505f9e20e3201d80b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.523ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{s}},}" loading="lazy"> such that the <a href="Denominator" class="mw-redirect" title="Denominator">denominator</a> s belongs to a given subset S of R. If S is the set of the non-zero elements of an <a href="Integral_domain" title="Integral domain">integral domain</a>, then the localization is the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a>: this case generalizes the construction of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

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</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> of <a href="Rational_number" title="Rational number">rational numbers</a> from the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> of <a href="Integer" title="Integer">integers</a>.

The technique has become fundamental, particularly in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, as it provides a natural link to <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> theory. In fact, the term localization originated in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>: if R is a ring of <a href="Function_(mathematics)" title="Function (mathematics)">functions</a> defined on some geometric object (<a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a>) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions that are not zero at p and localizes R with respect to S. The resulting ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> contains information about the behavior of V near p, and excludes information that is not "local", such as the <a href="Zero_of_a_function" title="Zero of a function">zeros of functions</a> that are outside V (cf. the example given at <a href="Local_ring" title="Local ring">local ring</a>).

<div class="mw-heading mw-heading2"><h2 id="Localization_of_a_ring">Localization of a ring</h2></div>

The localization of a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> R by a <a href="Multiplicatively_closed_set" title="Multiplicatively closed set">multiplicatively closed set</a> S is a new ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> whose elements are fractions with numerators in R and denominators in S.

If the ring is an <a href="Integral_domain" title="Integral domain">integral domain</a> the construction generalizes and follows closely that of the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a>, and, in particular, that of the <a href="Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a> as the field of fractions of the integers. For rings that have <a href="Zero_divisor" title="Zero divisor">zero divisors</a>, the construction is similar but requires more care.

<div class="mw-heading mw-heading3"><h3 id="Multiplicative_set">Multiplicative set</h3></div>

Localization is commonly done with respect to a <a href="Multiplicatively_closed_set" title="Multiplicatively closed set">multiplicatively closed set</a> S (also called a multiplicative set or a multiplicative system) of elements of a ring R, that is a subset of R that is <a href="Closure_(mathematics)" title="Closure (mathematics)">closed</a> under multiplication, and contains 1.

The requirement that S must be a multiplicative set is natural, since it implies that all denominators introduced by the localization belong to S. The localization by a set U that is not multiplicatively closed can also be defined, by taking as possible denominators all products of elements of U. However, the same localization is obtained by using the multiplicatively closed set S of all products of elements of U. As this often makes reasoning and notation simpler, it is standard practice to consider only localizations by multiplicative sets.

For example, the localization by a single element s introduces fractions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{s}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{s}},}</annotation>

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</math><img src="./0f2becb58b6b4d3620a17b7dafb05db37af5d34f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.353ex; height:3.009ex;" alt="{\displaystyle {\tfrac {a}{s}},}" loading="lazy"> but also products of such fractions, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {ab}{s^{2}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {ab}{s^{2}}}.}</annotation>

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</math><img src="./4e5f2aed936c0fbd5f31f92ada0ff47911b85ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:3.086ex; height:4.009ex;" alt="{\displaystyle {\tfrac {ab}{s^{2}}}.}" loading="lazy"> So, the denominators will belong to the multiplicative set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,s,s^{2},s^{3},\ldots \}}">

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<annotation encoding="application/x-tex">{\displaystyle \{1,s,s^{2},s^{3},\ldots \}}</annotation>

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</math><img src="./3551f2cb48e047d5cd82d8622ffebb830c127bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.726ex; height:3.176ex;" alt="{\displaystyle \{1,s,s^{2},s^{3},\ldots \}}" loading="lazy"> of the powers of s. Therefore, one generally talks of "the localization by the powers of an element" rather than of "the localization by an element".

The localization of a ring R by a multiplicative set S is generally denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R,}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R,}</annotation>

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</math><img src="./f8290ce316de84c0ba69323a19fa2092d313d575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.265ex; height:3.009ex;" alt="{\displaystyle S^{-1}R,}" loading="lazy"> but other notations are commonly used in some special cases: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{1,t,t^{2},\ldots \}}">

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<annotation encoding="application/x-tex">{\displaystyle S=\{1,t,t^{2},\ldots \}}</annotation>

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</math><img src="./08b05a821727f75ec99c816d3514a54cfc3aa7bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.644ex; height:3.176ex;" alt="{\displaystyle S=\{1,t,t^{2},\ldots \}}" loading="lazy"> consists of the powers of a single element, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is often denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{t};}">

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<annotation encoding="application/x-tex">{\displaystyle R_{t};}</annotation>

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</math><img src="./3b56c5c0a22c68f3732a721dd0e94f5896e2aa0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.509ex;" alt="{\displaystyle R_{t};}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R\setminus {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle S=R\setminus {\mathfrak {p}}}</annotation>

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</math><img src="./870ea9871818daf44541ec52a278d897acd0652a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.719ex; height:2.843ex;" alt="{\displaystyle S=R\setminus {\mathfrak {p}}}" loading="lazy"> is the <a href="Complement_(set_theory)" title="Complement (set theory)">complement</a> of a <a href="Prime_ideal" title="Prime ideal">prime ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

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</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}.}">

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<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}.}</annotation>

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</math><img src="./e531637e5073fa35a16704055918a0f2490e5281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.465ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}.}" loading="lazy">

In the remainder of this article, only localizations by a multiplicative set are considered.

<div class="mw-heading mw-heading3"><h3 id="Integral_domains">Integral domains</h3></div>

When the ring R is an <a href="Integral_domain" title="Integral domain">integral domain</a> and S does not contain 0, the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is a subring of the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of R. As such, the localization of a domain is a domain.

More precisely, it is the <a href="Subring" title="Subring">subring</a> of the field of fractions of R, that consists of the fractions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{s}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{s}}}</annotation>

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</math><img src="./4f77141122853c386a9b6051ef229271a05f6861.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.706ex; height:3.009ex;" alt="{\displaystyle {\tfrac {a}{s}}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S.}">

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<annotation encoding="application/x-tex">{\displaystyle s\in S.}</annotation>

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</math><img src="./28ab1cd4494f5bd54c4c6f5be29f5d6e17672f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.077ex; height:2.176ex;" alt="{\displaystyle s\in S.}" loading="lazy"> This is a subring since the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{s}}+{\tfrac {b}{t}}={\tfrac {at+bs}{st}},}">

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{s}}+{\tfrac {b}{t}}={\tfrac {at+bs}{st}},}</annotation>

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</math><img src="./f4a8a3d22a348f1cb1584903dc625de4d8fb7daa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.887ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a}{s}}+{\tfrac {b}{t}}={\tfrac {at+bs}{st}},}" loading="lazy"> and the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{s}}\,{\tfrac {b}{t}}={\tfrac {ab}{st}}}">

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<mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{s}}\,{\tfrac {b}{t}}={\tfrac {ab}{st}}}</annotation>

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</math><img src="./b65a7cb3f11c72932d6d1afec0d6cbc2465d165e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.144ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a}{s}}\,{\tfrac {b}{t}}={\tfrac {ab}{st}}}" loading="lazy"> of two elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> are in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R.}</annotation>

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</math><img src="./75fcba81243d001b630fa1a53361320df3255026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.265ex; height:2.676ex;" alt="{\displaystyle S^{-1}R.}" loading="lazy"> This results from the defining property of a multiplicative set, which implies also that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1={\tfrac {1}{1}}\in S^{-1}R.}">

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<annotation encoding="application/x-tex">{\displaystyle 1={\tfrac {1}{1}}\in S^{-1}R.}</annotation>

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</math><img src="./c49fe7beef3818997b022e87212bfaf0bb60ba31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.025ex; height:3.509ex;" alt="{\displaystyle 1={\tfrac {1}{1}}\in S^{-1}R.}" loading="lazy"> In this case, R is a subring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R.}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R.}</annotation>

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</math><img src="./75fcba81243d001b630fa1a53361320df3255026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.265ex; height:2.676ex;" alt="{\displaystyle S^{-1}R.}" loading="lazy"> It is shown below that this is no longer true in general, typically when S contains <a href="Zero_divisor" title="Zero divisor">zero divisors</a>.

For example, the <a href="Decimal_fraction" class="mw-redirect" title="Decimal fraction">decimal fractions</a> are the localization of the ring of integers by the multiplicative set of the powers of ten. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> consists of the rational numbers that can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {n}{10^{k}}},}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {n}{10^{k}}},}</annotation>

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</math><img src="./5190d257bfa37b9b16af54c2b3fe5dabb28dd4ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:3.986ex; height:3.676ex;" alt="{\displaystyle {\tfrac {n}{10^{k}}},}" loading="lazy"> where n is an integer, and k is a nonnegative integer.

<div class="mw-heading mw-heading3"><h3 id="General_construction">General construction</h3></div>

In the general case, a problem arises with <a href="Zero_divisor" title="Zero divisor">zero divisors</a>. Let S be a multiplicative set in a commutative ring R. Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S,}">

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<annotation encoding="application/x-tex">{\displaystyle s\in S,}</annotation>

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</math><img src="./9ad060d84dec285d2822233a418fbf98689f6ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.077ex; height:2.509ex;" alt="{\displaystyle s\in S,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq a\in R}">

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<annotation encoding="application/x-tex">{\displaystyle 0\neq a\in R}</annotation>

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</math><img src="./a6716ccac9b61f17ebc7879b5fbd138f32ed478a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.095ex; height:2.676ex;" alt="{\displaystyle 0\neq a\in R}" loading="lazy"> is a zero divisor with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle as=0.}">

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<annotation encoding="application/x-tex">{\displaystyle as=0.}</annotation>

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</math><img src="./59d288e2d50aaec698e1af25150fca71714850c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.228ex; height:2.176ex;" alt="{\displaystyle as=0.}" loading="lazy"> Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{1}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{1}}}</annotation>

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</math><img src="./dbe37bdbc904925d27dc28b5b4c31e2150a070c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.706ex; height:3.176ex;" alt="{\displaystyle {\tfrac {a}{1}}}" loading="lazy"> is the image in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in R,}">

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<annotation encoding="application/x-tex">{\displaystyle a\in R,}</annotation>

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</math><img src="./e60ad1d63f653a7249204f2f0f1330a542e0e581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.481ex; height:2.509ex;" alt="{\displaystyle a\in R,}" loading="lazy"> and one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{1}}={\tfrac {as}{s}}={\tfrac {0}{s}}={\tfrac {0}{1}}.}">

<mfrac>

</mfrac>

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<mfrac>

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<mfrac>

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<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{1}}={\tfrac {as}{s}}={\tfrac {0}{s}}={\tfrac {0}{1}}.}</annotation>

</semantics>

</math><img src="./577739ea3440ee0283ac8aa5e1c7f62b1e9ead8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.441ex; height:3.509ex;" alt="{\displaystyle {\tfrac {a}{1}}={\tfrac {as}{s}}={\tfrac {0}{s}}={\tfrac {0}{1}}.}" loading="lazy"> Thus some nonzero elements of R must be zero in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R.}</annotation>

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</math><img src="./75fcba81243d001b630fa1a53361320df3255026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.265ex; height:2.676ex;" alt="{\displaystyle S^{-1}R.}" loading="lazy"> The construction that follows is designed for taking this into account.

Given R and S as above, one considers the <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times S}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R\times S}</annotation>

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</math><img src="./55c3ff61b6cca09ae2b3fb47ba9417b51d83b94e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.104ex; height:2.176ex;" alt="{\displaystyle R\times S}" loading="lazy"> that is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2})}">

<msub>

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<msub>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2})}</annotation>

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</math><img src="./ee21454670670ea0ed7a9342c16f424d65b983a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.28ex; height:2.843ex;" alt="{\displaystyle (r_{1},s_{1})\sim (r_{2},s_{2})}" loading="lazy"> if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in S}</annotation>

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</math><img src="./c4e4de5cbd0086a33eceb5150ae9c19a73dde4be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.18ex; height:2.176ex;" alt="{\displaystyle t\in S}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}">

<msub>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}</annotation>

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</math><img src="./3ca164c23679a763a80324abe6498fe5bee9dcb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.892ex; height:2.843ex;" alt="{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}" loading="lazy">

The localization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is defined as the set of the <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> for this relation. The class of (r, s) is denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r}{s}},}">

<mfrac>

</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {r}{s}},}</annotation>

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</math><img src="./ff777e6715d380e7c148a74af2623efebefbb418.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.573ex; height:4.676ex;" alt="{\displaystyle {\frac {r}{s}},}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r/s,}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle r/s,}</annotation>

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</math><img src="./3a4813c2de575c1a6186eb77dba149154db5d980.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.948ex; height:2.843ex;" alt="{\displaystyle r/s,}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{-1}r.}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle s^{-1}r.}</annotation>

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</math><img src="./9feebf2a5abf378dd7ecbddc1b88d496e0f79f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.119ex; height:2.676ex;" alt="{\displaystyle s^{-1}r.}" loading="lazy"> So, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {r_{1}}{s_{1}}}={\tfrac {r_{2}}{s_{2}}}}">

<mfrac>

<msub>

</mrow>

</msub>

<msub>

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</mfrac>

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</mrow>

<mfrac>

<msub>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {r_{1}}{s_{1}}}={\tfrac {r_{2}}{s_{2}}}}</annotation>

</semantics>

</math><img src="./0a1b0ed0b2ea58039327784cf638253e94325184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.976ex; height:3.676ex;" alt="{\displaystyle {\tfrac {r_{1}}{s_{1}}}={\tfrac {r_{2}}{s_{2}}}}" loading="lazy"> if and only if there is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in S}</annotation>

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</math><img src="./c4e4de5cbd0086a33eceb5150ae9c19a73dde4be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.18ex; height:2.176ex;" alt="{\displaystyle t\in S}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}">

<msub>

</mrow>

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<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}</annotation>

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</math><img src="./3ca164c23679a763a80324abe6498fe5bee9dcb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.892ex; height:2.843ex;" alt="{\displaystyle t(s_{1}r_{2}-s_{2}r_{1})=0.}" loading="lazy"> The reason for the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> is to handle cases such as the above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a}{1}}={\tfrac {0}{1}},}">

<mfrac>

</mfrac>

</mstyle>

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<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {a}{1}}={\tfrac {0}{1}},}</annotation>

</semantics>

</math><img src="./c2608619a2a6e50439dc3129cc5fd552b32da85f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.109ex; height:3.509ex;" alt="{\displaystyle {\tfrac {a}{1}}={\tfrac {0}{1}},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1}r_{2}-s_{2}r_{1}}">

<msub>

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</msub>

<msub>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle s_{1}r_{2}-s_{2}r_{1}}</annotation>

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</math><img src="./9a1081a4dce69b6e26306cbc50108ecc7294e5bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.336ex; height:2.343ex;" alt="{\displaystyle s_{1}r_{2}-s_{2}r_{1}}" loading="lazy"> is nonzero even though the fractions should be regarded as equal.

The localization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is a commutative ring with addition

<mfrac>

<msub>

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</mfrac>

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<mfrac>

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</mfrac>

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<mfrac>

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<mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {r_{1}}{s_{1}}}+{\frac {r_{2}}{s_{2}}}={\frac {r_{1}s_{2}+r_{2}s_{1}}{s_{1}s_{2}}},}</annotation>

</semantics>

</math><img src="./76fb1bc50d74dde78dab239cd6ed3130630be47f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.719ex; height:5.343ex;" alt="{\displaystyle {\frac {r_{1}}{s_{1}}}+{\frac {r_{2}}{s_{2}}}={\frac {r_{1}s_{2}+r_{2}s_{1}}{s_{1}s_{2}}},}" loading="lazy"></dd></dl>

multiplication

<mfrac>

<msub>

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<mfrac>

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</msub>

<msub>

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</mfrac>

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<mfrac>

<mrow>

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</msub>

<msub>

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<mrow>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {r_{1}}{s_{1}}}\,{\frac {r_{2}}{s_{2}}}={\frac {r_{1}r_{2}}{s_{1}s_{2}}},}</annotation>

</semantics>

</math><img src="./f61b52459275528d9a0492e15b5c5d5348305db4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.22ex; height:5.009ex;" alt="{\displaystyle {\frac {r_{1}}{s_{1}}}\,{\frac {r_{2}}{s_{2}}}={\frac {r_{1}r_{2}}{s_{1}s_{2}}},}" loading="lazy"></dd></dl>

<a href="Additive_identity" title="Additive identity">additive identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {0}{1}},}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {0}{1}},}</annotation>

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</math><img src="./215d048b2c38d3299eb60cc89b87d2f55f39beab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.305ex; height:3.509ex;" alt="{\displaystyle {\tfrac {0}{1}},}" loading="lazy"> and <a href="Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{1}}.}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{1}}.}</annotation>

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</math><img src="./9abf3b60fb04a4d773ec010635d686339ee100af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.305ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{1}}.}" loading="lazy">

The <a href="Function_(mathematics)" title="Function (mathematics)">function</a>

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle r\mapsto {\frac {r}{1}}}</annotation>

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</math><img src="./37816a91b1121b2ebcd72b57cf1cdca919d1e896.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.661ex; height:4.676ex;" alt="{\displaystyle r\mapsto {\frac {r}{1}}}" loading="lazy"></dd></dl>

defines a <a href="Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R,}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R,}</annotation>

</semantics>

</math><img src="./f8290ce316de84c0ba69323a19fa2092d313d575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.265ex; height:3.009ex;" alt="{\displaystyle S^{-1}R,}" loading="lazy"> which is <a href="Injective_function" title="Injective function">injective</a> if and only if S does not contain any zero divisors.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\in S,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\in S,}</annotation>

</semantics>

</math><img src="./4c0e2a135a19219579298ed5dcf5657cbef5c06c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.149ex; height:2.509ex;" alt="{\displaystyle 0\in S,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is the <a href="Zero_ring" title="Zero ring">zero ring</a> that has only one unique element 0.

If S is the set of all <a href="Zero_divisor" title="Zero divisor">regular elements</a> of R (that is the elements that are not zero divisors), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is called the <a href="Total_ring_of_fractions" title="Total ring of fractions">total ring of fractions</a> of R.

<div class="mw-heading mw-heading3"><h3 id="Universal_property">Universal property</h3></div>

The (above defined) ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon R\to S^{-1}R}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle j\colon R\to S^{-1}R}</annotation>

</semantics>

</math><img src="./25515c0f52f035215f16d9e2f701069d8bcd2e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:13.015ex; height:3.009ex;" alt="{\displaystyle j\colon R\to S^{-1}R}" loading="lazy"> satisfies a <a href="Universal_property" title="Universal property">universal property</a> that is described below. This characterizes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

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</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> up to an <a href="Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">isomorphism</a>. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be more technical.

The universal property satisfied by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon R\to S^{-1}R}">

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<annotation encoding="application/x-tex">{\displaystyle j\colon R\to S^{-1}R}</annotation>

</semantics>

</math><img src="./25515c0f52f035215f16d9e2f701069d8bcd2e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:13.015ex; height:3.009ex;" alt="{\displaystyle j\colon R\to S^{-1}R}" loading="lazy"> is the following:

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon R\to T}</annotation>

</semantics>

</math><img src="./286fe74d9144fe3c9546838df70eb8982dad4b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.327ex; height:2.509ex;" alt="{\displaystyle f\colon R\to T}" loading="lazy"> is a ring homomorphism that maps every element of S to a <a href="Unit_(ring_theory)" title="Unit (ring theory)">unit</a> (invertible element) in T, there exists a unique ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon S^{-1}R\to T}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\colon S^{-1}R\to T}</annotation>

</semantics>

</math><img src="./b452ccbbc21f0f36386b8e479001e84b6cc3325c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.019ex; height:3.009ex;" alt="{\displaystyle g\colon S^{-1}R\to T}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g\circ j.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=g\circ j.}</annotation>

</semantics>

</math><img src="./bbf64dcb3d82775ec6e4f0e22ea18cc0650bdb8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.293ex; height:2.509ex;" alt="{\displaystyle f=g\circ j.}" loading="lazy"></dd></dl>

Using <a href="Category_theory" title="Category theory">category theory</a>, this can be expressed by saying that localization is a <a href="Functor" title="Functor">functor</a> that is <a href="Left_adjoint" class="mw-redirect" title="Left adjoint">left adjoint</a> to a <a href="Forgetful_functor" title="Forgetful functor">forgetful functor</a>. More precisely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

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</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {D}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation>

</semantics>

</math><img src="./3277962e1959c3241fb1b70c7f0ac6dcefebd966.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}" loading="lazy"> be the categories whose objects are <a href="Ordered_pair" title="Ordered pair">pairs</a> of a commutative ring and a <a href="Submonoid" class="mw-redirect" title="Submonoid">submonoid</a> of, respectively, the multiplicative <a href="Monoid" title="Monoid">monoid</a> or the <a href="Group_of_units" class="mw-redirect" title="Group of units">group of units</a> of the ring. The <a href="Morphism" title="Morphism">morphisms</a> of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\colon {\mathcal {D}}\to {\mathcal {C}}}">

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\colon {\mathcal {D}}\to {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./cb6f7fe609f165f311fdb597ccb12cf2cf924dd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.606ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}\colon {\mathcal {D}}\to {\mathcal {C}}}" loading="lazy"> be the forgetful functor that forgets that the elements of the second element of the pair are invertible.

Then the factorization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g\circ j}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f=g\circ j}</annotation>

</semantics>

</math><img src="./eef76945ec10937f76ce3704e78e11882b2515bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.646ex; height:2.509ex;" alt="{\displaystyle f=g\circ j}" loading="lazy"> of the universal property defines a bijection

<msub>

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<msub>

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</msub>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \hom _{\mathcal {C}}((R,S),{\mathcal {F}}(T,U))\to \hom _{\mathcal {D}}((S^{-1}R,j(S)),(T,U)).}</annotation>

</semantics>

</math><img src="./2518885d39d2b91e6715af945791e10c6ea854db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.622ex; height:3.176ex;" alt="{\displaystyle \hom _{\mathcal {C}}((R,S),{\mathcal {F}}(T,U))\to \hom _{\mathcal {D}}((S^{-1}R,j(S)),(T,U)).}" loading="lazy"></dd></dl>

This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R=\mathbb {Z} }</annotation>

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</math><img src="./c7d60fc180e76c47f663220cdb9ff8c6f564c0f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.413ex; height:2.176ex;" alt="{\displaystyle R=\mathbb {Z} }" loading="lazy"> is the ring of <a href="Integer" title="Integer">integers</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\mathbb {Z} \setminus \{0\},}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\mathbb {Z} \setminus \{0\},}</annotation>

</semantics>

</math><img src="./209b309e3f097295d95415f8e0bde3da8f4f1a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.477ex; height:2.843ex;" alt="{\displaystyle S=\mathbb {Z} \setminus \{0\},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation>

</semantics>

</math><img src="./c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> of the <a href="Rational_number" title="Rational number">rational numbers</a>.</li>

<li>If R is an <a href="Integral_domain" title="Integral domain">integral domain</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R\setminus \{0\},}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=R\setminus \{0\},}</annotation>

</semantics>

</math><img src="./c1efba720039a77ad3c856965f7eabcbef42d5e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.691ex; height:2.843ex;" alt="{\displaystyle S=R\setminus \{0\},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of R. The preceding example is a special case of this one.</li>

<li>If R is a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>, and if S is the subset of its elements that are not <a href="Zero_divisor" title="Zero divisor">zero divisors</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is the <a href="Total_ring_of_fractions" title="Total ring of fractions">total ring of fractions</a> of R. In this case, S is the largest multiplicative set such that the homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\to S^{-1}R}</annotation>

</semantics>

</math><img src="./86f6b81ba73abd9e60efc82671f1d237706ef61f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.997ex; height:2.676ex;" alt="{\displaystyle R\to S^{-1}R}" loading="lazy"> is injective. The preceding example is a special case of this one.</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is an element of a commutative ring R and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{1,x,x^{2},\ldots \},}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\{1,x,x^{2},\ldots \},}</annotation>

</semantics>

</math><img src="./a497c46c586aa8078a1af756141e4256a62e3df8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.271ex; height:3.176ex;" alt="{\displaystyle S=\{1,x,x^{2},\ldots \},}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> can be identified (is <a href="Canonical_isomorphism" class="mw-redirect" title="Canonical isomorphism">canonically isomorphic</a> to) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[x^{-1}]=R[s]/(xs-1).}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R[x^{-1}]=R[s]/(xs-1).}</annotation>

</semantics>

</math><img src="./e48a9da765bfc2e8f6b4128755b950eaa05ddb66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.008ex; height:3.176ex;" alt="{\displaystyle R[x^{-1}]=R[s]/(xs-1).}" loading="lazy"> (The proof consists of showing that this ring satisfies the above universal property.) The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is generally denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{x}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{x}}</annotation>

</semantics>

</math><img src="./997624be443bc4f27f01ab5bd96c7a700f6d5868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.936ex; height:2.509ex;" alt="{\displaystyle R_{x}}" loading="lazy">.<a href="#cite_note-1">[1]</a> This sort of localization plays a fundamental role in the definition of an <a href="Affine_scheme" class="mw-redirect" title="Affine scheme">affine scheme</a>.</li>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> is a <a href="Prime_ideal" title="Prime ideal">prime ideal</a> of a commutative ring R, the <a href="Set_complement" class="mw-redirect" title="Set complement">set complement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R\setminus {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=R\setminus {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./870ea9871818daf44541ec52a278d897acd0652a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.719ex; height:2.843ex;" alt="{\displaystyle S=R\setminus {\mathfrak {p}}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> in R is a multiplicative set (by the definition of a prime ideal). The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is a <a href="Local_ring" title="Local ring">local ring</a> that is generally denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}},}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}},}</annotation>

</semantics>

</math><img src="./df0ebba61aa0d10e00667e4cca185c8b7edeb6c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.465ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}},}" loading="lazy"> and called the local ring of R at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}.}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}.}</annotation>

</semantics>

</math><img src="./47c2a3f50fca87e9d975ab518d50201690df35ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}.}" loading="lazy"> This sort of localization is fundamental in <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a>, because many properties of a commutative ring can be read on its local rings. Such a property is often called a <a href="Local_property" title="Local property">local property</a>. For example, a ring is <a href="Regular_ring" class="mw-redirect" title="Regular ring">regular</a> if and only if all its local rings are regular.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Ring_properties">Ring properties</h3></div>

Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a>, <a href="Module_(mathematics)" title="Module (mathematics)">modules</a>, or several multiplicative sets are considered in other sections.

<msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R=0}</annotation>

</semantics>

</math><img src="./59de39838757fd933869d902421a737dd5129316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.879ex; height:2.676ex;" alt="{\displaystyle S^{-1}R=0}" loading="lazy"> <a href="If_and_only_if" title="If and only if">if and only if</a> S contains 0.</li>

<li>The <a href="Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to S^{-1}R}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle R\to S^{-1}R}</annotation>

</semantics>

</math><img src="./86f6b81ba73abd9e60efc82671f1d237706ef61f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.997ex; height:2.676ex;" alt="{\displaystyle R\to S^{-1}R}" loading="lazy"> is injective if and only if S does not contain any <a href="Zero_divisor" title="Zero divisor">zero divisors</a>.</li>

<li>The ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to S^{-1}R}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R\to S^{-1}R}</annotation>

</semantics>

</math><img src="./86f6b81ba73abd9e60efc82671f1d237706ef61f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.997ex; height:2.676ex;" alt="{\displaystyle R\to S^{-1}R}" loading="lazy"> is an <a href="Epimorphism" title="Epimorphism">epimorphism</a> in the <a href="Category_of_rings" title="Category of rings">category of rings</a>, that is not <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> in general.</li>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is a <a href="Flat_module" title="Flat module">flat R-module</a> (see <a href="#Localization_of_a_module">§ Localization of a module</a> for details).</li>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle S=R\setminus {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./870ea9871818daf44541ec52a278d897acd0652a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.719ex; height:2.843ex;" alt="{\displaystyle S=R\setminus {\mathfrak {p}}}" loading="lazy"> is the <a href="Complement_(set_theory)" title="Complement (set theory)">complement</a> of a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R,}">

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle S^{-1}R,}</annotation>

</semantics>

</math><img src="./f8290ce316de84c0ba69323a19fa2092d313d575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.265ex; height:3.009ex;" alt="{\displaystyle S^{-1}R,}" loading="lazy"> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}},}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}},}</annotation>

</semantics>

</math><img src="./df0ebba61aa0d10e00667e4cca185c8b7edeb6c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.465ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}},}" loading="lazy"> is a <a href="Local_ring" title="Local ring">local ring</a>; that is, it has only one <a href="Maximal_ideal" title="Maximal ideal">maximal ideal</a>.</li>

<li>Localization commutes with formations of finite sums, products, intersections and radicals;<a href="#cite_note-2">[2]</a> e.g., if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {I}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {I}}}</annotation>

</semantics>

</math><img src="./84930d16ca4b945cc8d757cac5691bf82fbb2895.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.108ex; height:3.009ex;" alt="{\displaystyle {\sqrt {I}}}" loading="lazy"> denote the <a href="Radical_of_an_ideal" title="Radical of an ideal">radical of an ideal</a> I in R, then</li></ul>

<msqrt>

</msqrt>

</mrow>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {I}}\cdot S^{-1}R={\sqrt {I\cdot S^{-1}R}}\,.}</annotation>

</semantics>

</math><img src="./998d4a55671797decd78ab300dce3c29929caf5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.331ex; height:3.343ex;" alt="{\displaystyle {\sqrt {I}}\cdot S^{-1}R={\sqrt {I\cdot S^{-1}R}}\,.}" loading="lazy"></dd></dl></dd>

<dd>In particular, R is <a href="Reduced_ring" title="Reduced ring">reduced</a> if and only if its total ring of fractions is reduced.<a href="#cite_note-3">[3]</a></dd></dl>

<ul><li>Let R be an integral domain with the field of fractions K. Then its localization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./124aee5f6d80492d2746e652dee942c36ef6e1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}" loading="lazy"> at a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> can be viewed as a subring of K. Moreover,</li></ul>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=\bigcap _{\mathfrak {p}}R_{\mathfrak {p}}=\bigcap _{\mathfrak {m}}R_{\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./c6cd62cb0054906b01e3a52fcd0f2e29a0e2b3e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.973ex; height:5.843ex;" alt="{\displaystyle R=\bigcap _{\mathfrak {p}}R_{\mathfrak {p}}=\bigcap _{\mathfrak {m}}R_{\mathfrak {m}}}" loading="lazy"></dd></dl></dd>

<dd>where the first intersection is over all prime ideals and the second over the maximal ideals.<a href="#cite_note-4">[4]</a></dd></dl>

<ul><li>There is a <a href="Bijection" title="Bijection">bijection</a> between the set of prime ideals of S−1R and the set of prime ideals of R that are <a href="Disjoint_sets" title="Disjoint sets">disjoint</a> from S. This bijection is induced by the given homomorphism R → S −1R.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Saturation_of_a_multiplicative_set">Saturation of a multiplicative set</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq R}</annotation>

</semantics>

</math><img src="./b9a9719c1e3e0e1ba37e28af70f70851bdd099a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.362ex; height:2.343ex;" alt="{\displaystyle S\subseteq R}" loading="lazy"> be a multiplicative set. The saturation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}}</annotation>

</semantics>

</math><img src="./84040a3e50cb11792bdb6cfaac286b46476e3447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.573ex; height:2.843ex;" alt="{\displaystyle {\hat {S}}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is the set

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}=\{r\in R\colon \exists s\in R,rs\in S\}.}</annotation>

</semantics>

</math><img src="./bb3d4009e70d4a0b09a86fc521c0a32e61c7a001.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.831ex; height:3.343ex;" alt="{\displaystyle {\hat {S}}=\{r\in R\colon \exists s\in R,rs\in S\}.}" loading="lazy"></dd></dl>

The multiplicative set S is saturated if it equals its saturation, that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}=S}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}=S}</annotation>

</semantics>

</math><img src="./a3e1fd49c1c965a3c233a27cd951ffa967777afb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.17ex; height:2.843ex;" alt="{\displaystyle {\hat {S}}=S}" loading="lazy">, or equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rs\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle rs\in S}</annotation>

</semantics>

</math><img src="./1b3406335afc131cbf231f9838dd3f39b59e5a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.479ex; height:2.176ex;" alt="{\displaystyle rs\in S}" loading="lazy"> implies that r and s are in S.

If S is not saturated, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rs\in S,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle rs\in S,}</annotation>

</semantics>

</math><img src="./2247066feb76fad1ffa74eb8108a041566529dce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.126ex; height:2.509ex;" alt="{\displaystyle rs\in S,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {s}{rs}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {s}{rs}}}</annotation>

</semantics>

</math><img src="./a2298de4d4b353e0cf93c4adf285fcc092038ed0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.975ex; height:4.676ex;" alt="{\displaystyle {\frac {s}{rs}}}" loading="lazy"> is a <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of the image of r in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R.}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R.}</annotation>

</semantics>

</math><img src="./75fcba81243d001b630fa1a53361320df3255026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.265ex; height:2.676ex;" alt="{\displaystyle S^{-1}R.}" loading="lazy"> So, the images of the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}}</annotation>

</semantics>

</math><img src="./84040a3e50cb11792bdb6cfaac286b46476e3447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.573ex; height:2.843ex;" alt="{\displaystyle {\hat {S}}}" loading="lazy"> are all invertible in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R,}</annotation>

</semantics>

</math><img src="./f8290ce316de84c0ba69323a19fa2092d313d575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.265ex; height:3.009ex;" alt="{\displaystyle S^{-1}R,}" loading="lazy"> and the universal property implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {S}}{}^{-1}R}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {S}}{}^{-1}R}</annotation>

</semantics>

</math><img src="./c1cc35a250b4fa456d76b5671d5736dd54057aff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.669ex; height:2.843ex;" alt="{\displaystyle {\hat {S}}{}^{-1}R}" loading="lazy"> are <a href="Canonical_isomorphism" class="mw-redirect" title="Canonical isomorphism">canonically isomorphic</a>, that is, there is a unique isomorphism between them that fixes the images of the elements of R.

If S and T are two multiplicative sets, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T^{-1}R}</annotation>

</semantics>

</math><img src="./855812595e4853c3ab87f7dae56849080f05fdde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.817ex; height:2.676ex;" alt="{\displaystyle T^{-1}R}" loading="lazy"> are isomorphic if and only if they have the same saturation, or, equivalently, if s belongs to one of the multiplicative sets, then there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in R}</annotation>

</semantics>

</math><img src="./3c9f9e17cef0b56a955ded4bf99ca09cd14417f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.444ex; height:2.176ex;" alt="{\displaystyle t\in R}" loading="lazy"> such that st belongs to the other.

Saturated multiplicative sets are not widely used explicitly, since, for verifying that a set is saturated, one must know all <a href="Unit_(ring_theory)" title="Unit (ring theory)">units</a> of the ring.

<div class="mw-heading mw-heading2"><h2 id="Terminology_explained_by_the_context">Terminology explained by the context</h2></div>

The term localization originates in the general trend of modern mathematics to study <a href="Geometry" title="Geometry">geometrical</a> and <a href="Topology" title="Topology">topological</a> objects locally, that is in terms of their behavior near each point. Examples of this trend are the fundamental concepts of <a href="Manifold" title="Manifold">manifolds</a>, <a href="Germ_(mathematics)" title="Germ (mathematics)">germs</a> and <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheafs</a>. In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, an <a href="Affine_algebraic_set" class="mw-redirect" title="Affine algebraic set">affine algebraic set</a> can be identified with a <a href="Quotient_ring" title="Quotient ring">quotient ring</a> of a <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> in such a way that the points of the algebraic set correspond to the <a href="Maximal_ideal" title="Maximal ideal">maximal ideals</a> of the ring (this is <a href="Hilbert's_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>). This correspondence has been generalized for making the set of the <a href="Prime_ideal" title="Prime ideal">prime ideals</a> of a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> a <a href="Topological_space" title="Topological space">topological space</a> equipped with the <a href="Zariski_topology" title="Zariski topology">Zariski topology</a>; this topological space is called the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of the ring</a>.

In this context, a localization by a multiplicative set may be viewed as the restriction of the spectrum of a ring to the subspace of the prime ideals (viewed as points) that do not intersect the multiplicative set.

Two classes of localizations are more commonly considered:

<ul><li>The multiplicative set is the <a href="Complement_(set_theory)" title="Complement (set theory)">complement</a> of a <a href="Prime_ideal" title="Prime ideal">prime ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> of a ring R. In this case, one speaks of the "localization at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">", or "localization at a point". The resulting ring, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./124aee5f6d80492d2746e652dee942c36ef6e1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}" loading="lazy"> is a <a href="Local_ring" title="Local ring">local ring</a>, and is the algebraic analog of a <a href="Germ_(mathematics)#Ring_of_germs" title="Germ (mathematics)">ring of germs</a>.</li>

<li>The multiplicative set consists of all powers of an element t of a ring R. The resulting ring is commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{t},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{t},}</annotation>

</semantics>

</math><img src="./d1bceabf782c596e2534c13a56391d43edc3d31a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.237ex; height:2.509ex;" alt="{\displaystyle R_{t},}" loading="lazy"> and its spectrum is the Zariski open set of the prime ideals that do not contain t. Thus the localization is the analog of the restriction of a topological space to a neighborhood of a point (every prime ideal has a <a href="Neighborhood_basis" class="mw-redirect" title="Neighborhood basis">neighborhood basis</a> consisting of Zariski open sets of this form).</li></ul>

In <a href="Number_theory" title="Number theory">number theory</a> and <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a>, when working over the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> of <a href="Integer" title="Integer">integers</a>, one refers to a property relative to an integer n as a property true at n or away from n, depending on the localization that is considered. "Away from n" means that the property is considered after localization by the powers of n, and, if p is a <a href="Prime_number" title="Prime number">prime number</a>, "at p" means that the property is considered after localization at the prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\mathbb {Z} }</annotation>

</semantics>

</math><img src="./f28bf16b40b47b6da6f2649085c5dde23fb24e3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.809ex; height:2.509ex;" alt="{\displaystyle p\mathbb {Z} }" loading="lazy">. This terminology can be explained by the fact that, if p is prime, the nonzero prime ideals of the localization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> are either the <a href="Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> {p} or its complement in the set of prime numbers.

<div class="mw-heading mw-heading2"><h2 id="Localization_and_saturation_of_ideals">Localization and saturation of ideals</h2></div>

Let S be a multiplicative set in a commutative ring R, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\colon R\to S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j\colon R\to S^{-1}R}</annotation>

</semantics>

</math><img src="./25515c0f52f035215f16d9e2f701069d8bcd2e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:13.015ex; height:3.009ex;" alt="{\displaystyle j\colon R\to S^{-1}R}" loading="lazy"> be the canonical ring homomorphism. Given an <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> I in R, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}I}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}I}</annotation>

</semantics>

</math><img src="./8e306a3e19edb87156f7249ec297017ad3a28c8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.026ex; height:2.676ex;" alt="{\displaystyle S^{-1}I}" loading="lazy"> the set of the fractions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> whose numerator is in I. This is an ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R,}</annotation>

</semantics>

</math><img src="./f8290ce316de84c0ba69323a19fa2092d313d575.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.265ex; height:3.009ex;" alt="{\displaystyle S^{-1}R,}" loading="lazy"> which is generated by j(I), and called the localization of I by S.

The saturation of I by S is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{-1}(S^{-1}I);}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j^{-1}(S^{-1}I);}</annotation>

</semantics>

</math><img src="./d16accca8ec7065cf3ed0e6887b5e6872165913e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:10.8ex; height:3.176ex;" alt="{\displaystyle j^{-1}(S^{-1}I);}" loading="lazy"> it is an ideal of R, which can also defined as the set of the elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation>

</semantics>

</math><img src="./ca49c66b5e9b5f32249a737e4429c3df136c33f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}" loading="lazy"> such that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation>

</semantics>

</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle sr\in I.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle sr\in I.}</annotation>

</semantics>

</math><img src="./475c9503aba82c7fb4d1ca37f87bc10650a5075e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.798ex; height:2.176ex;" alt="{\displaystyle sr\in I.}" loading="lazy">

Many properties of ideals are either preserved by saturation and localization, or can be characterized by simpler properties of localization and saturation.

In what follows, S is a multiplicative set in a ring R, and I and J are ideals of R; the saturation of an ideal I by a multiplicative set S is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sat} _{S}(I),}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sat} _{S}(I),}</annotation>

</semantics>

</math><img src="./4abbd2bd9680f8cc5aef3c29c231de84f8035555.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.904ex; height:2.843ex;" alt="{\displaystyle \operatorname {sat} _{S}(I),}" loading="lazy"> or, when the multiplicative set S is clear from the context, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sat} (I).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sat} (I).}</annotation>

</semantics>

</math><img src="./2debe0664c2e0557d47a73f633f9c9d02bd59391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.611ex; height:2.843ex;" alt="{\displaystyle \operatorname {sat} (I).}" loading="lazy">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\in S^{-1}I\quad \iff \quad 1\in \operatorname {sat} (I)\quad \iff \quad S\cap I\neq \emptyset }</annotation>

</semantics>

</math><img src="./e9ca4fbcce93afb057013d4dd379b71cc2eba595.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.596ex; height:3.176ex;" alt="{\displaystyle 1\in S^{-1}I\quad \iff \quad 1\in \operatorname {sat} (I)\quad \iff \quad S\cap I\neq \emptyset }" loading="lazy"></li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\subseteq J\quad \ \implies \quad \ S^{-1}I\subseteq S^{-1}J\quad \ {\text{and}}\quad \ \operatorname {sat} (I)\subseteq \operatorname {sat} (J)}</annotation>

</semantics>

</math><img src="./4eceba063fc7c8682a5d5694524f4e9e9daed866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.653ex; height:3.176ex;" alt="{\displaystyle I\subseteq J\quad \ \implies \quad \ S^{-1}I\subseteq S^{-1}J\quad \ {\text{and}}\quad \ \operatorname {sat} (I)\subseteq \operatorname {sat} (J)}" loading="lazy"> (this is not always true for <a href="Strict_subset" class="mw-redirect" title="Strict subset">strict inclusions</a>)</li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}(I\cap J)=S^{-1}I\cap S^{-1}J,\qquad \,\operatorname {sat} (I\cap J)=\operatorname {sat} (I)\cap \operatorname {sat} (J)}</annotation>

</semantics>

</math><img src="./3eb854d2ff2d1e40219aa4ec626cfab0e3b08b05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.916ex; height:3.176ex;" alt="{\displaystyle S^{-1}(I\cap J)=S^{-1}I\cap S^{-1}J,\qquad \,\operatorname {sat} (I\cap J)=\operatorname {sat} (I)\cap \operatorname {sat} (J)}" loading="lazy"></li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}(I+J)=S^{-1}I+S^{-1}J,\qquad \operatorname {sat} (I+J)=\operatorname {sat} (I)+\operatorname {sat} (J)}</annotation>

</semantics>

</math><img src="./152ccdde49bff44a922e82563a456e23eeaeb80f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.561ex; height:3.176ex;" alt="{\displaystyle S^{-1}(I+J)=S^{-1}I+S^{-1}J,\qquad \operatorname {sat} (I+J)=\operatorname {sat} (I)+\operatorname {sat} (J)}" loading="lazy"></li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}(I\cdot J)=S^{-1}I\cdot S^{-1}J,\qquad \quad \operatorname {sat} (I\cdot J)=\operatorname {sat} (I)\cdot \operatorname {sat} (J)}</annotation>

</semantics>

</math><img src="./47c1ea8117de24726bf961ad7d1da1e525e2e11e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.238ex; height:3.176ex;" alt="{\displaystyle S^{-1}(I\cdot J)=S^{-1}I\cdot S^{-1}J,\qquad \quad \operatorname {sat} (I\cdot J)=\operatorname {sat} (I)\cdot \operatorname {sat} (J)}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> is a <a href="Prime_ideal" title="Prime ideal">prime ideal</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\cap S=\emptyset ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\cap S=\emptyset ,}</annotation>

</semantics>

</math><img src="./50066b821071f79877a9bd161d8d5047dfd2c84d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.152ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}\cap S=\emptyset ,}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}{\mathfrak {p}}}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./f8b1a8b8b4e8d33277b0f40ad585ae54a0f1d3ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.017ex; height:3.009ex;" alt="{\displaystyle S^{-1}{\mathfrak {p}}}" loading="lazy"> is a prime ideal and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}=\operatorname {sat} ({\mathfrak {p}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}=\operatorname {sat} ({\mathfrak {p}})}</annotation>

</semantics>

</math><img src="./c9e1c10de237a6de18039cffa795ce450640e19a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.216ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}=\operatorname {sat} ({\mathfrak {p}})}" loading="lazy">; if the intersection is nonempty, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}{\mathfrak {p}}=S^{-1}R}">

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}{\mathfrak {p}}=S^{-1}R}</annotation>

</semantics>

</math><img src="./7d6925d1842f16b329f45ae8bac10f26286fe873.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.734ex; height:3.009ex;" alt="{\displaystyle S^{-1}{\mathfrak {p}}=S^{-1}R}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sat} ({\mathfrak {p}})=R.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sat} ({\mathfrak {p}})=R.}</annotation>

</semantics>

</math><img src="./cdfa34aa5ecfa26cedb62c1d85bd259ba66445f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.464ex; height:2.843ex;" alt="{\displaystyle \operatorname {sat} ({\mathfrak {p}})=R.}" loading="lazy"></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Localization_of_a_module">Localization of a module</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> be a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> be a <a href="Multiplicative_set" class="mw-redirect" title="Multiplicative set">multiplicative set</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> be an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-<a href="Module_(mathematics)" title="Module (mathematics)">module</a>. The localization of the module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}M}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}M}</annotation>

</semantics>

</math><img src="./970c76a4224771f7e43ec2b337f781a2265273bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.297ex; height:2.676ex;" alt="{\displaystyle S^{-1}M}" loading="lazy">, is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy">-module that is constructed exactly as the localization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, except that the numerators of the fractions belong to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. That is, as a set, it consists of <a href="Equivalence_class" title="Equivalence class">equivalence classes</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {m}{s}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {m}{s}}}</annotation>

</semantics>

</math><img src="./799b101a4f13b6f6688abba97f0a37084a94fd13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.877ex; height:4.676ex;" alt="{\displaystyle {\frac {m}{s}}}" loading="lazy">, of pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,s)}</annotation>

</semantics>

</math><img src="./147a28dcd22db3774c5689c105135a16f440880b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.974ex; height:2.843ex;" alt="{\displaystyle (m,s)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\in M}</annotation>

</semantics>

</math><img src="./0c5790f9afa538086b9fea356114e77099c1a775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.176ex;" alt="{\displaystyle m\in M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in S,}</annotation>

</semantics>

</math><img src="./9ad060d84dec285d2822233a418fbf98689f6ddb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.077ex; height:2.509ex;" alt="{\displaystyle s\in S,}" loading="lazy"> and two pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m,s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m,s)}</annotation>

</semantics>

</math><img src="./147a28dcd22db3774c5689c105135a16f440880b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.974ex; height:2.843ex;" alt="{\displaystyle (m,s)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n,t)}</annotation>

</semantics>

</math><img src="./2c3ec72aa185c61b6ecd8d858c4f4c520b940fd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle (n,t)}" loading="lazy"> are equivalent if there is an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> such that

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(sn-tm)=0.}</annotation>

</semantics>

</math><img src="./2c1833aa1ff137028869d95ba49572d27b588a72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.252ex; height:2.843ex;" alt="{\displaystyle u(sn-tm)=0.}" loading="lazy"></dd></dl>

Addition and scalar multiplication are defined as for usual fractions (in the following formula, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in R,}</annotation>

</semantics>

</math><img src="./552e90ad66ce281fa21e771e83f192364fe1ffc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.3ex; height:2.509ex;" alt="{\displaystyle r\in R,}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,t\in S,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,t\in S,}</annotation>

</semantics>

</math><img src="./ea6395df181004b344434c9cef0ac49140f17a0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.951ex; height:2.509ex;" alt="{\displaystyle s,t\in S,}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,n\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,n\in M}</annotation>

</semantics>

</math><img src="./1746e683f508f8e51c4f9f829e209bcd4860e511.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.752ex; height:2.509ex;" alt="{\displaystyle m,n\in M}" loading="lazy">):

<mfrac>

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<mfrac>

</mfrac>

</mrow>

<mfrac>

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</mrow>

<mrow>

</mrow>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {m}{s}}+{\frac {n}{t}}={\frac {tm+sn}{st}},}</annotation>

</semantics>

</math><img src="./dd18e6af724ba0b1892db119859582c06cd7c68c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.735ex; height:5.176ex;" alt="{\displaystyle {\frac {m}{s}}+{\frac {n}{t}}={\frac {tm+sn}{st}},}" loading="lazy"></dd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

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</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {r}{s}}{\frac {m}{t}}={\frac {rm}{st}}.}</annotation>

</semantics>

</math><img src="./84346619f649081d7802b0162033a38f6bbdc87b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.474ex; height:4.676ex;" alt="{\displaystyle {\frac {r}{s}}{\frac {m}{t}}={\frac {rm}{st}}.}" loading="lazy"></dd></dl>

Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}M}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}M}</annotation>

</semantics>

</math><img src="./970c76a4224771f7e43ec2b337f781a2265273bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.297ex; height:2.676ex;" alt="{\displaystyle S^{-1}M}" loading="lazy"> is also an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">-module with scalar multiplication

<mfrac>

</mfrac>

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<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\,{\frac {m}{s}}={\frac {r}{1}}{\frac {m}{s}}={\frac {rm}{s}}.}</annotation>

</semantics>

</math><img src="./ddc7bbe56316de2cfce366c67699dbe92bee572f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.956ex; height:4.676ex;" alt="{\displaystyle r\,{\frac {m}{s}}={\frac {r}{1}}{\frac {m}{s}}={\frac {rm}{s}}.}" loading="lazy"></dd></dl>

It is straightforward to check that these operations are well-defined, that is, they give the same result for different choices of representatives of fractions.

The localization of a module can be equivalently defined by using <a href="Tensor_product_of_modules" title="Tensor product of modules">tensor products</a>:

<msup>

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</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}M=S^{-1}R\otimes _{R}M.}</annotation>

</semantics>

</math><img src="./a68262f0c396ebba0319ccaef946d0e7c43b4da9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.423ex; height:3.009ex;" alt="{\displaystyle S^{-1}M=S^{-1}R\otimes _{R}M.}" loading="lazy"></dd></dl>

The proof of equivalence (up to a <a href="Canonical_isomorphism" class="mw-redirect" title="Canonical isomorphism">canonical isomorphism</a>) can be done by showing that the two definitions satisfy the same universal property.

<div class="mw-heading mw-heading3"><h3 id="Module_properties">Module properties</h3></div>

If M is a <a href="Submodule" class="mw-redirect" title="Submodule">submodule</a> of an R-module N, and S is a multiplicative set in R, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}M\subseteq S^{-1}N.}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}M\subseteq S^{-1}N.}</annotation>

</semantics>

</math><img src="./13f496e67132393e2870acf4d9a0bf8c8a75af02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.96ex; height:2.843ex;" alt="{\displaystyle S^{-1}M\subseteq S^{-1}N.}" loading="lazy"> This implies that, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>

</semantics>

</math><img src="./26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> is an <a href="Injective" class="mw-redirect" title="Injective">injective</a> <a href="Module_homomorphism" title="Module homomorphism">module homomorphism</a>, then

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R\otimes _{R}f:\quad S^{-1}R\otimes _{R}M\to S^{-1}R\otimes _{R}N}</annotation>

</semantics>

</math><img src="./be69faf4daa84560c3e0a4d896bf53aad5d8bb7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.474ex; height:3.009ex;" alt="{\displaystyle S^{-1}R\otimes _{R}f:\quad S^{-1}R\otimes _{R}M\to S^{-1}R\otimes _{R}N}" loading="lazy"></dd></dl>

is also an injective homomorphism.

Since the tensor product is a <a href="Right_exact_functor" class="mw-redirect" title="Right exact functor">right exact functor</a>, this implies that localization by S maps <a href="Exact_sequence" title="Exact sequence">exact sequences</a> of R-modules to exact sequences of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy">-modules. In other words, localization is an <a href="Exact_functor" title="Exact functor">exact functor</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is a <a href="Flat_module" title="Flat module">flat R-module</a>.

This flatness and the fact that localization solves a <a href="Universal_property" title="Universal property">universal property</a> make that localization preserves many properties of modules and rings, and is compatible with solutions of other universal properties. For example, the <a href="Natural_transformation" title="Natural transformation">natural map</a>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

<msup>

</mrow>

</msup>

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</msub>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}(M\otimes _{R}N)\to S^{-1}M\otimes _{S^{-1}R}S^{-1}N}</annotation>

</semantics>

</math><img src="./515d1750bf035e479a73388200f3124565cc10bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.584ex; height:3.343ex;" alt="{\displaystyle S^{-1}(M\otimes _{R}N)\to S^{-1}M\otimes _{S^{-1}R}S^{-1}N}" loading="lazy"></dd></dl>

is an isomorphism. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a <a href="Finitely_presented_module" class="mw-redirect" title="Finitely presented module">finitely presented module</a>, the natural map

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}\operatorname {Hom} _{R}(M,N)\to \operatorname {Hom} _{S^{-1}R}(S^{-1}M,S^{-1}N)}</annotation>

</semantics>

</math><img src="./030679956e3f5b46df13b9bfc9b49f6b04165372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.85ex; height:3.343ex;" alt="{\displaystyle S^{-1}\operatorname {Hom} _{R}(M,N)\to \operatorname {Hom} _{S^{-1}R}(S^{-1}M,S^{-1}N)}" loading="lazy"></dd></dl>

is also an isomorphism.<a href="#cite_note-5">[5]</a>

If a module M is a <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> over R, one has

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

<msup>

</mrow>

</msup>

</mrow>

</msub>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}(\operatorname {Ann} _{R}(M))=\operatorname {Ann} _{S^{-1}R}(S^{-1}M),}</annotation>

</semantics>

</math><img src="./2a1e49b195f5c8e3da27d8a840865e1b15495da2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.328ex; height:3.343ex;" alt="{\displaystyle S^{-1}(\operatorname {Ann} _{R}(M))=\operatorname {Ann} _{S^{-1}R}(S^{-1}M),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ann} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ann} }</annotation>

</semantics>

</math><img src="./2dc9e175cde9bd4fe9cab3f2c879c26673d57d58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.328ex; height:2.176ex;" alt="{\displaystyle \operatorname {Ann} }" loading="lazy"> denotes <a href="Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a>, that is the ideal of the elements of the ring that map to zero all elements of the module.<a href="#cite_note-6">[6]</a> In particular,

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}M=0\quad \iff \quad S\cap \operatorname {Ann} _{R}(M)\neq \emptyset ,}</annotation>

</semantics>

</math><img src="./06022ea5ae3d9ff736bc91489dc78d35a00e0dd1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.149ex; height:3.176ex;" alt="{\displaystyle S^{-1}M=0\quad \iff \quad S\cap \operatorname {Ann} _{R}(M)\neq \emptyset ,}" loading="lazy"></dd></dl>

that is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle tM=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle tM=0}</annotation>

</semantics>

</math><img src="./b610f94e0023b00e70c4284b7976fe1cf84b5e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.543ex; height:2.176ex;" alt="{\displaystyle tM=0}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in S.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\in S.}</annotation>

</semantics>

</math><img src="./b12231f39e621de20cfe25c11d6a3fde9f231d74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.826ex; height:2.176ex;" alt="{\displaystyle t\in S.}" loading="lazy"><a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Localization_at_primes">Localization at primes</h2></div>

The definition of a <a href="Prime_ideal" title="Prime ideal">prime ideal</a> implies immediately that the <a href="Set_complement" class="mw-redirect" title="Set complement">complement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R\setminus {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=R\setminus {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./870ea9871818daf44541ec52a278d897acd0652a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.719ex; height:2.843ex;" alt="{\displaystyle S=R\setminus {\mathfrak {p}}}" loading="lazy"> of a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> in a commutative ring R is a multiplicative set. In this case, the localization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}R}</annotation>

</semantics>

</math><img src="./36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> is commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}.}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}.}</annotation>

</semantics>

</math><img src="./e531637e5073fa35a16704055918a0f2490e5281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.465ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}.}" loading="lazy"> The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./124aee5f6d80492d2746e652dee942c36ef6e1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}" loading="lazy"> is a <a href="Local_ring" title="Local ring">local ring</a>, that is called the local ring of R at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}.}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}.}</annotation>

</semantics>

</math><img src="./47c2a3f50fca87e9d975ab518d50201690df35ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}.}" loading="lazy"> This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\,R_{\mathfrak {p}}={\mathfrak {p}}\otimes _{R}R_{\mathfrak {p}}}">

</mrow>

<msub>

</mrow>

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</mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\,R_{\mathfrak {p}}={\mathfrak {p}}\otimes _{R}R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./e27097fe963577953c34675c8fffea41896d9c6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.767ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}\,R_{\mathfrak {p}}={\mathfrak {p}}\otimes _{R}R_{\mathfrak {p}}}" loading="lazy"> is the unique <a href="Maximal_ideal" title="Maximal ideal">maximal ideal</a> of the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}.}">

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}.}</annotation>

</semantics>

</math><img src="./e531637e5073fa35a16704055918a0f2490e5281.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.465ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}.}" loading="lazy"> Analogously one can define the localization of a module M at a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> of R. Again, the localization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}M}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{-1}M}</annotation>

</semantics>

</math><img src="./970c76a4224771f7e43ec2b337f781a2265273bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.297ex; height:2.676ex;" alt="{\displaystyle S^{-1}M}" loading="lazy"> is commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathfrak {p}}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./ca7387604e6a21fb117d96e11706b02683856ce5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.308ex; height:2.843ex;" alt="{\displaystyle M_{\mathfrak {p}}}" loading="lazy">.

Such localizations are fundamental for commutative algebra and algebraic geometry for several reasons. One is that local rings are often easier to study than general commutative rings, in particular because of <a href="Nakayama_lemma" class="mw-redirect" title="Nakayama lemma">Nakayama lemma</a>. However, the main reason is that many properties are true for a ring if and only if they are true for all its local rings. For example, a ring is <a href="Regular_ring" class="mw-redirect" title="Regular ring">regular</a> if and only if all its local rings are <a href="Regular_local_ring" title="Regular local ring">regular local rings</a>.

Properties of a ring that can be characterized on its local rings are called local properties, and are often the algebraic counterpart of geometric <a href="Local_property" title="Local property">local properties</a> of <a href="Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a>, which are properties that can be studied by restriction to a small neighborhood of each point of the variety. (There is another concept of local property that refers to localization to Zariski open sets; see <a href="#Localization_to_Zariski_open_sets">§ Localization to Zariski open sets</a>, below.)

Many local properties are a consequence of the fact that the module

</mrow>

</munder>

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{\mathfrak {p}}R_{\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./8ecdfaf5b2c3544daee8ee1f28cc176f11219a6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:6.716ex; height:5.843ex;" alt="{\displaystyle \bigoplus _{\mathfrak {p}}R_{\mathfrak {p}}}" loading="lazy"></dd></dl>

is a <a href="Faithfully_flat_module" class="mw-redirect" title="Faithfully flat module">faithfully flat module</a> when the direct sum is taken over all prime ideals (or over all <a href="Maximal_ideal" title="Maximal ideal">maximal ideals</a> of R). See also <a href="Faithfully_flat_descent" title="Faithfully flat descent">Faithfully flat descent</a>.

<div class="mw-heading mw-heading3"><h3 id="Examples_of_local_properties">Examples of local properties</h3></div>

A property P of an R-module M is a local property if the following conditions are equivalent:

<ul><li>P holds for M.</li>

<li>P holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathfrak {p}},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathfrak {p}},}</annotation>

</semantics>

</math><img src="./5bd202a2abdb266206993a3044b87db289731fc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.955ex; height:2.843ex;" alt="{\displaystyle M_{\mathfrak {p}},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation>

</semantics>

</math><img src="./a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> is a prime ideal of R.</li>

<li>P holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathfrak {m}},}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathfrak {m}},}</annotation>

</semantics>

</math><img src="./2cbe7c62866180c6c0b7c75b5eee38c23bce01c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.394ex; height:2.509ex;" alt="{\displaystyle M_{\mathfrak {m}},}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation>

</semantics>

</math><img src="./adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> is a maximal ideal of R.</li></ul>

The following are local properties:

<li>M is torsion-free (in the case where R is a <a href="Commutative_domain" class="mw-redirect" title="Commutative domain">commutative domain</a>).</li>

<li>M is a <a href="Flat_module" title="Flat module">flat module</a>.</li>

<li>M is an <a href="Invertible_module" title="Invertible module">invertible module</a> (in the case where R is a commutative domain, and M is a submodule of the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> of R).</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M\to N}</annotation>

</semantics>

</math><img src="./26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> is injective (resp. surjective), where N is another R-module.</li></ul>

On the other hand, some properties are not local properties. For example, an infinite <a href="Direct_product" title="Direct product">direct product</a> of <a href="Field_(mathematics)" title="Field (mathematics)">fields</a> is not an <a href="Integral_domain" title="Integral domain">integral domain</a> nor a <a href="Noetherian_ring" title="Noetherian ring">Noetherian ring</a>, while all its local rings are fields, and therefore Noetherian integral domains.

<div class="mw-heading mw-heading2"><h2 id="Non-commutative_case">Non-commutative case</h2></div>

Localizing <a href="Non-commutative_ring" class="mw-redirect" title="Non-commutative ring">non-commutative rings</a> is more difficult. While the localization exists for every set S of prospective units, it might take a different form to the one described above. One condition which ensures that the localization is well behaved is the <a href="Ore_condition" title="Ore condition">Ore condition</a>.

One case for non-commutative rings where localization has a clear interest is for rings of <a href="Differential_operators" class="mw-redirect" title="Differential operators">differential operators</a>. It has the interpretation, for example, of adjoining a formal inverse D−1 for a differentiation operator D. This is done in many contexts in methods for <a href="Differential_equation" title="Differential equation">differential equations</a>. There is now a large mathematical theory about it, named <a href="Microlocal_analysis" title="Microlocal analysis">microlocalization</a>, connecting with numerous other branches. The micro- tag is to do with connections with <a href="Fourier_theory" class="mw-redirect" title="Fourier theory">Fourier theory</a>, in particular.

<ul><li><a href="Local_analysis" title="Local analysis">Local analysis</a></li>

<li><a href="Localization_of_a_category" title="Localization of a category">Localization of a category</a></li>

<li><a href="Localization_of_a_topological_space" title="Localization of a topological space">Localization of a topological space</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> This definition makes sense even if x is <a href="Nilpotent" title="Nilpotent">nilpotent</a>, which would make S a finite set that contains 0, but in that case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{x}=S^{-1}R=0}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{x}=S^{-1}R=0}</annotation>

</semantics>

</math><img src="./58ef5dd99a103d5a30f7b8a5a7939946e54019c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.914ex; height:3.009ex;" alt="{\displaystyle R_{x}=S^{-1}R=0}" loading="lazy">.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFAtiyahMacdonald1969">Atiyah & Macdonald 1969</a>, Proposition 3.11. (v).

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Borel, AG. 3.3

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Matsumura, Theorem 4.7

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFEisenbud1995">Eisenbud 1995</a>, Proposition 2.10

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFAtiyahMacdonald1969">Atiyah & Macdonald 1969</a>, Proposition 3.14.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Borel, AG. 3.1

</li>

</ol></div></div>

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</style><cite id="CITEREFAtiyahMacdonald1969" class="citation book cs1"><a href="Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael Francis</a>; <a href="Ian_G._Macdonald" title="Ian G. Macdonald">Macdonald, I.G.</a> (1969). Introduction to Commutative Algebra. Westview Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-40751-8</bdi>.</cite></li>

<li><a href="Armand_Borel" title="Armand Borel">Borel, Armand</a>. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-97370-2</bdi>.</li>

<li><cite id="CITEREFCohn1989" class="citation book cs1">Cohn, P. M. (1989). "§ 9.3". Algebra. Vol. 2 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xvi+428. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-92234-X</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1006872">1006872</a>.</cite></li>

<li><cite id="CITEREFCohn1991" class="citation book cs1">Cohn, P. M. (1991). "§ 9.1". Algebra. Vol. 3 (2nd ed.). Chichester: John Wiley & Sons Ltd. pp. xii+474. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-92840-2</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1098018">1098018</a>.</cite></li>

<li><cite id="CITEREFEisenbud1995" class="citation cs2"><a href="David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995), Commutative algebra, Graduate Texts in Mathematics, vol. 150, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94268-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a></cite></li>

<li><cite id="CITEREFMatsumura1980" class="citation book cs1"><a href="Hideyuki_Matsumura" title="Hideyuki Matsumura">Matsumura, Hideyuki</a> (1980). Commutative Algebra. Mathematics Lecture Note Series. Vol. 56 (2nd ed.). Reading, Massachusetts: The Benjamin/Cummings Publishing Company Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-7026-9</bdi>.</cite></li>

<li><cite id="CITEREFStenström1971" class="citation book cs1">Stenström, Bo (1971). Rings and modules of quotients. Lecture Notes in Mathematics, Vol. 237. Berlin: Springer-Verlag. pp. vii+136. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-05690-4</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0325663">0325663</a>.</cite></li>

<li><a href="Serge_Lang" title="Serge Lang">Serge Lang</a>, "Algebraic Number Theory," Springer, 2000. pages 3–4.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Localization.html">Localization</a> from <a href="MathWorld" title="MathWorld">MathWorld</a>.</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-08" href="https://en.wikipedia.org/wiki/?title=Localization_(commutative_algebra)&oldid=1304767007">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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