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Local ring

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In <a href="Mathematics" title="Mathematics">mathematics</a>, more specifically in <a href="Ring_theory" title="Ring theory">ring theory</a>, local rings are certain <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a> that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on <a href="Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a> or <a href="Manifold" title="Manifold">manifolds</a>, or of <a href="Algebraic_number_fields" class="mw-redirect" title="Algebraic number fields">algebraic number fields</a> examined at a particular <a href="Place_(mathematics)" class="mw-redirect" title="Place (mathematics)">place</a>, or prime. Local algebra is the branch of <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a> that studies <a href="Commutative_ring" title="Commutative ring">commutative</a> local rings and their <a href="Module_(mathematics)" title="Module (mathematics)">modules</a>.

In practice, a commutative local ring often arises as the result of the <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization of a ring</a> at a <a href="Prime_ideal" title="Prime ideal">prime ideal</a>.

The concept of local rings was introduced by <a href="Wolfgang_Krull" title="Wolfgang Krull">Wolfgang Krull</a> in 1938 under the name Stellenringe.<a href="#cite_note-Krull-1">[1]</a> The English term local ring is due to <a href="Zariski" class="mw-redirect" title="Zariski">Zariski</a>.<a href="#cite_note-Zariski-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition_and_first_consequences">Definition and first consequences</h2></div>

A <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> R is a local ring if it has any one of the following equivalent properties:

<ul><li>R has a unique <a href="Maximal_ideal" title="Maximal ideal">maximal</a> left <a href="Ring_ideal" class="mw-redirect" title="Ring ideal">ideal</a>.</li>

<li>R has a unique maximal right ideal.</li>

<li>1 ≠ 0 and the sum of any two non-<a href="Unit_(algebra)" class="mw-redirect" title="Unit (algebra)">units</a> in R is a non-unit.</li>

<li>1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.</li>

<li>If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).</li></ul>

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's <a href="Jacobson_radical" title="Jacobson radical">Jacobson radical</a>. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,<a href="#cite_note-3">[3]</a> necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a> proper (<a href="Principal_ideal" title="Principal ideal">principal</a>) (left) ideals, where two ideals I1, I2 are called coprime if R = I1 + I2.

In the case of <a href="Commutative_ring" title="Commutative ring">commutative rings</a>, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.

Before about 1960 many authors required that a local ring be (left and right) <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a>, and (possibly non-Noetherian) local rings were called quasi-local rings. In this article this requirement is not imposed.

A local ring that is an <a href="Integral_domain" title="Integral domain">integral domain</a> is called a local domain.

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

<ul><li>All <a href="Field_(mathematics)" title="Field (mathematics)">fields</a> (and <a href="Skew_field" class="mw-redirect" title="Skew field">skew fields</a>) are local rings, since {0} is the only maximal ideal in these rings.</li>

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

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</math><img src="./3f568eec3d80f812c3d02a19a445f088e888c180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.651ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }" loading="lazy"> is a local ring (p prime, n ≥ 1). The unique maximal ideal consists of all multiples of p.</li>

<li>More generally, a nonzero ring in which every element is either a unit or <a href="Nilpotent" title="Nilpotent">nilpotent</a> is a local ring.</li>

<li>An important class of local rings are <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation rings</a>, which are local <a href="Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a> that are not fields.</li>

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

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</math><img src="./a73c34ee2333f14e92e4dc6c5f8da064bd9ff211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.595ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [[x]]}" loading="lazy">, whose elements are infinite series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=0}^{\infty }a_{i}x^{i}}">

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<annotation encoding="application/x-tex">{\textstyle \sum _{i=0}^{\infty }a_{i}x^{i}}</annotation>

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</math><img src="./d367e97d01e136187d29083a365de1cb77a42b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.9ex; height:3.176ex;" alt="{\textstyle \sum _{i=0}^{\infty }a_{i}x^{i}}" loading="lazy"> where multiplications are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\sum _{i=0}^{\infty }a_{i}x^{i})(\sum _{i=0}^{\infty }b_{i}x^{i})=\sum _{i=0}^{\infty }c_{i}x^{i}}">

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<annotation encoding="application/x-tex">{\textstyle (\sum _{i=0}^{\infty }a_{i}x^{i})(\sum _{i=0}^{\infty }b_{i}x^{i})=\sum _{i=0}^{\infty }c_{i}x^{i}}</annotation>

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</math><img src="./8a707779d937c7e93b02f00ca80de9ee08f29ea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.962ex; height:3.176ex;" alt="{\textstyle (\sum _{i=0}^{\infty }a_{i}x^{i})(\sum _{i=0}^{\infty }b_{i}x^{i})=\sum _{i=0}^{\infty }c_{i}x^{i}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c_{n}=\sum _{i+j=n}a_{i}b_{j}}">

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<annotation encoding="application/x-tex">{\textstyle c_{n}=\sum _{i+j=n}a_{i}b_{j}}</annotation>

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</math><img src="./359d53a3ba1be152ef77853de5f128a4037eff0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.122ex; height:3.343ex;" alt="{\textstyle c_{n}=\sum _{i+j=n}a_{i}b_{j}}" loading="lazy">, is local. Its unique maximal ideal consists of all elements that are not invertible. In other words, it consists of all elements with constant term zero.</li>

<li>More generally, every ring of <a href="Formal_power_series" title="Formal power series">formal power series</a> over a local ring is local; the maximal ideal consists of those power series with <a href="Constant_term" title="Constant term">constant term</a> in the maximal ideal of the base ring.</li>

<li>Similarly, the <a href="Algebra_over_a_field" title="Algebra over a field">algebra</a> of <a href="Dual_numbers" class="mw-redirect" title="Dual numbers">dual numbers</a> over any field is local. More generally, if F is a local ring and n is a positive integer, then the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> F[X]/(Xn) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of F, since one can use a <a href="Geometric_series" title="Geometric series">geometric series</a> to invert all other polynomials <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">modulo</a> Xn. If F is a field, then elements of F[X]/(Xn) are either <a href="Nilpotent" title="Nilpotent">nilpotent</a> or <a href="Invertible" class="mw-redirect" title="Invertible">invertible</a>. (The dual numbers over F correspond to the case n = 2.)</li>

<li>Nonzero quotient rings of local rings are local.</li>

<li>The ring of <a href="Rational_number" title="Rational number">rational numbers</a> with <a href="Odd_number" class="mw-redirect" title="Odd number">odd</a> denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integers <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localized</a> at 2.</li>

<li>More generally, given any <a href="Commutative_ring" title="Commutative ring">commutative ring</a> R and any <a href="Prime_ideal" title="Prime ideal">prime ideal</a> P of R, the <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization</a> of R at P is local; the maximal ideal is the ideal generated by P in this localization; that is, the maximal ideal consists of all elements a/s with a ∈ P and s ∈ R - P.</li></ul>

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

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<ul><li>The <a href="Polynomial_ring" title="Polynomial ring">ring of polynomials</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K[x]}">

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

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

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is not local, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-x}">

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

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<li>The ring of integers <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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<annotation encoding="application/x-tex">{\displaystyle (p)}</annotation>

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

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

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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">, where p and q are distinct prime numbers. Both (p) and (q) are maximal ideals here.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Ring_of_germs">Ring of germs</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Germ_(mathematics)" title="Germ (mathematics)">Germ (mathematics)</a></div>

To motivate the name "local" for these rings, we consider real-valued <a href="Continuous_function" title="Continuous function">continuous functions</a> defined on some <a href="Interval_(mathematics)" title="Interval (mathematics)">open interval</a> around 0 of the <a href="Real_line" class="mw-redirect" title="Real line">real line</a>. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>, and the <a href="Equivalence_class" title="Equivalence class">equivalence classes</a> are what are called the "<a href="Germ_(mathematics)" title="Germ (mathematics)">germs</a> of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.

To see that this ring of germs is local, we need to characterize its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then by continuity there is an open interval around 0 where f is non-zero, and we can form the function g(x) = 1/f(x) on this interval. The function g gives rise to a germ, and the product of fg is equal to 1. (Conversely, if f is invertible, then there is some g such that f(0)g(0) = 1, hence f(0) ≠ 0.)

With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs f with f(0) = 0.

Exactly the same arguments work for the ring of germs of continuous real-valued functions on any <a href="Topological_space" title="Topological space">topological space</a> at a given point, or the ring of germs of <a href="Differentiable" class="mw-redirect" title="Differentiable">differentiable</a> functions on any <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a> at a given point, or the ring of germs of <a href="Rational_function" title="Rational function">rational functions</a> on any <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a> at a given point. All these rings are therefore local. These examples help to explain why <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a>, the generalizations of varieties, are defined as special <a href="Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed spaces</a>.

<div class="mw-heading mw-heading3"><h3 id="Valuation_theory">Valuation theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Valuation_(algebra)" title="Valuation (algebra)">Valuation (algebra)</a></div>

Local rings play a major role in valuation theory. By definition, a <a href="Valuation_ring" title="Valuation ring">valuation ring</a> of a field K is a subring R such that for every non-zero element x of K, at least one of x and x−1 is in R. Any such subring will be a local ring. For example, the ring of <a href="Rational_number" title="Rational number">rational numbers</a> with <a href="Odd_number" class="mw-redirect" title="Odd number">odd</a> denominator (mentioned above) is a valuation ring in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }">

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Given a field K, which may or may not be a <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a>, we may look for local rings in it. If K were indeed the function field of an <a href="Algebraic_variety" title="Algebraic variety">algebraic variety</a> V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or more there is a difficulty that is seen this way: if F and G are rational functions on V with

the function

is an <a href="Indeterminate_form" title="Indeterminate form">indeterminate form</a> at P. Considering a simple example, such as

approached along a line

one sees that the value at P is a concept without a simple definition. It is replaced by using valuations.

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

Non-commutative local rings arise naturally as <a href="Endomorphism_ring" title="Endomorphism ring">endomorphism rings</a> in the study of <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> decompositions of <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a>; conversely, if the module M has finite <a href="Length_of_a_module" title="Length of a module">length</a> and is indecomposable, then its endomorphism ring is local.

If k is a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> p > 0 and G is a finite <a href="P-group" title="P-group">p-group</a>, then the <a href="Group_ring" title="Group ring">group algebra</a> kG is local.

<div class="mw-heading mw-heading2"><h2 id="Some_facts_and_definitions">Some facts and definitions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Commutative_case">Commutative case</h3></div>

We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a <a href="Topological_ring" title="Topological ring">topological ring</a> in a natural way if one takes the powers of m as a <a href="Neighborhood_base" class="mw-redirect" title="Neighborhood base">neighborhood base</a> of 0. This is the <a href="I-adic_topology" title="I-adic topology">m-adic topology</a> on R. If (R, m) is a commutative <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a> local ring, then

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<annotation encoding="application/x-tex">{\displaystyle \bigcap _{i=1}^{\infty }m^{i}=\{0\}}</annotation>

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</math><img src="./d9aaa19f6e18fe926a4d50716a1afa9eb9e33fdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.481ex; height:6.843ex;" alt="{\displaystyle \bigcap _{i=1}^{\infty }m^{i}=\{0\}}" loading="lazy"></dd></dl>

(Krull's intersection theorem), and it follows that R with the m-adic topology is a <a href="Hausdorff_space" title="Hausdorff space">Hausdorff space</a>. The theorem is a consequence of the <a href="Artin%E2%80%93Rees_lemma" title="Artin–Rees lemma">Artin–Rees lemma</a> together with <a href="Nakayama's_lemma" title="Nakayama's lemma">Nakayama's lemma</a>, and, as such, the "Noetherian" assumption is crucial. Indeed, let R be the ring of germs of infinitely differentiable functions at 0 in the real line and m be the maximal ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dd1f6d437f1742bfd5ffbdccd2477c07e9909e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle (x)}" loading="lazy">. Then a nonzero function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{-{1 \over x^{2}}}}">

<msup>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{-{1 \over x^{2}}}}</annotation>

</semantics>

</math><img src="./17fe5afc61706dee77a4c49551ce78ac31ae75c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.994ex; height:4.009ex;" alt="{\displaystyle e^{-{1 \over x^{2}}}}" loading="lazy"> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a4b18ef9f5eba3b2ed06a6bc7c546cda76e702f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.259ex; height:2.343ex;" alt="{\displaystyle m^{n}}" loading="lazy"> for any n, since that function divided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./150d38e238991bc4d0689ffc9d2a852547d2658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}" loading="lazy"> is still smooth.

As for any topological ring, one can ask whether (R, m) is <a href="Complete_uniform_space" class="mw-redirect" title="Complete uniform space">complete</a> (as a <a href="Uniform_space" title="Uniform space">uniform space</a>); if it is not, one considers its <a href="Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">completion</a>, again a local ring. Complete Noetherian local rings are classified by the <a href="Cohen_structure_theorem" title="Cohen structure theorem">Cohen structure theorem</a>.

In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, especially when R is the local ring of a scheme at some point P, R / m is called the <a href="Residue_field" title="Residue field">residue field</a> of the local ring or residue field of the point P.

If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a <a href="Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> f : R → S with the property f(m) ⊆ n.<a href="#cite_note-4">[4]</a> These are precisely the ring homomorphisms that are continuous with respect to the given topologies on R and S. For example, consider the ring morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x,y]/(x^{3},x^{2}y,y^{4})}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x,y]/(x^{3},x^{2}y,y^{4})}</annotation>

</semantics>

</math><img src="./3796f3f56429806e8065c2c158c5aa4228245125.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.94ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x,y]/(x^{3},x^{2}y,y^{4})}" loading="lazy"> sending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto x}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./033c0ae81eaf4c65cbb0759d7aa2c4f434c00f02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.273ex; height:1.843ex;" alt="{\displaystyle x\mapsto x}" loading="lazy">. The preimage of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./41cf50e4a314ca8e2c30964baa8d26e5be7a9386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle (x,y)}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dd1f6d437f1742bfd5ffbdccd2477c07e9909e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle (x)}" loading="lazy">. Another example of a local ring morphism is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x]/(x^{2})}">

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x]/(x^{2})}</annotation>

</semantics>

</math><img src="./727a191de27018976a83ace720491ad075a313d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.928ex; height:3.176ex;" alt="{\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x]/(x^{2})}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="General_case">General case</h3></div>

The <a href="Jacobson_radical" title="Jacobson radical">Jacobson radical</a> m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.<a href="#cite_note-5">[5]</a>

For an element x of the local ring R, the following are equivalent:

<ul><li>x has a left inverse</li>

<li>x has a right inverse</li>

<li>x is invertible</li>

If (R, m) is local, then the <a href="Factor_ring" class="mw-redirect" title="Factor ring">factor ring</a> R/m is a <a href="Skew_field" class="mw-redirect" title="Skew field">skew field</a>. If J ≠ R is any two-sided ideal in R, then the factor ring R/J is again local, with maximal ideal m/J.

A <a href="Kaplansky's_theorem_on_projective_modules" title="Kaplansky's theorem on projective modules">deep theorem</a> by <a href="Irving_Kaplansky" title="Irving Kaplansky">Irving Kaplansky</a> says that any <a href="Projective_module" title="Projective module">projective module</a> over a local ring is <a href="Free_module" title="Free module">free</a>, though the case where the module is finitely-generated is a simple corollary to <a href="Nakayama's_lemma" title="Nakayama's lemma">Nakayama's lemma</a>. This has an interesting consequence in terms of <a href="Morita_equivalence" title="Morita equivalence">Morita equivalence</a>. Namely, if P is a <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> projective R module, then P is isomorphic to the free module Rn, and hence the ring of endomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} _{R}(P)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {End} _{R}(P)}</annotation>

</semantics>

</math><img src="./e0f562e54064a9533627cdd18722e3c194548533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.202ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} _{R}(P)}" loading="lazy"> is isomorphic to the full ring of matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {M} _{n}(R)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {M} _{n}(R)}</annotation>

</semantics>

</math><img src="./6e83703a58173a1c28a596b5eb397e73442c5e53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.923ex; height:2.843ex;" alt="{\displaystyle \mathrm {M} _{n}(R)}" loading="lazy">. Since every ring Morita equivalent to the local ring R is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {End} _{R}(P)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {End} _{R}(P)}</annotation>

</semantics>

</math><img src="./e0f562e54064a9533627cdd18722e3c194548533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.202ex; height:2.843ex;" alt="{\displaystyle \mathrm {End} _{R}(P)}" loading="lazy"> for such a P, the conclusion is that the only rings Morita equivalent to a local ring R are (isomorphic to) the matrix rings over R.

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

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFKrull1938" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Wolfgang_Krull" title="Wolfgang Krull">Krull, Wolfgang</a> (1938). "Dimensionstheorie in Stellenringen". J. Reine Angew. Math. (in German). 1938 (179): 204. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1938.179.204">10.1515/crll.1938.179.204</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115691729">115691729</a>.</cite>

</li>

<cite id="CITEREFZariski1943" class="citation journal cs1"><a href="Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (May 1943). <a rel="nofollow" class="external text" href="http://www.ams.org/tran/1943-053-03/S0002-9947-1943-0008468-9/S0002-9947-1943-0008468-9.pdf">"Foundations of a General Theory of Birational Correspondences"</a> (PDF). Trans. Amer. Math. Soc. 53 (3). American Mathematical Society: 490–542 [497]. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990215">10.2307/1990215</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990215">1990215</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Lam (2001), p. 295, Thm. 19.1.

</li>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> The 2 by 2 matrices over a field, for example, has unique maximal ideal {0}, but it has multiple maximal right and left ideals.

</li>

</ol></div>

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

<ul><li><cite id="CITEREFLam2001" class="citation book cs1"><a href="T.Y._Lam" class="mw-redirect" title="T.Y. Lam">Lam, T.Y.</a> (2001). A first course in noncommutative rings. Graduate Texts in Mathematics (2nd ed.). Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95183-0</bdi>.</cite></li>

<li><cite id="CITEREFJacobson2009" class="citation book cs1"><a href="Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009). Basic algebra. Vol. 2 (2nd ed.). Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-47187-7</bdi>.</cite></li></ul>

<ul><li><a href="Discrete_valuation_ring" title="Discrete valuation ring">Discrete valuation ring</a></li>

<li><a href="Semi-local_ring" title="Semi-local ring">Semi-local ring</a></li>

<li><a href="Gorenstein_local_ring" class="mw-redirect" title="Gorenstein local ring">Gorenstein local ring</a></li>

<li><a href="Regular_local_ring" title="Regular local ring">Regular local ring</a></li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/255511">The philosophy behind local rings</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-01" href="https://en.wikipedia.org/wiki/?title=Local_ring&oldid=1293456304">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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