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Free module

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In <a href="Mathematics" title="Mathematics">mathematics</a>, a free module is a <a href="Module_(mathematics)" title="Module (mathematics)">module</a> that has a basis, that is, a <a href="Generating_set_of_a_module" title="Generating set of a module">generating set</a> that is <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a>. Every <a href="Vector_space" title="Vector space">vector space</a> is a free module,<a href="#cite_note-1">[1]</a> but, if the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of the coefficients is not a <a href="Division_ring" title="Division ring">division ring</a> (not a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> in the <a href="Commutative_ring" title="Commutative ring">commutative</a> case), then there exist non-free modules.

Given any <a href="Set_(mathematics)" title="Set (mathematics)">set</a> S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.

A <a href="Free_abelian_group" title="Free abelian group">free abelian group</a> is precisely a free module over 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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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

For a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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

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

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

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

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</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">, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subseteq M}">

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

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</math><img src="./5fbe408f36b55e9249974a3c3282f666c5d85764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.316ex; height:2.343ex;" alt="{\displaystyle E\subseteq M}" loading="lazy"> is a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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"> if:

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

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is a <a href="Generating_set_of_a_module" title="Generating set of a module">generating set</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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 to say, every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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 finite sum of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> multiplied by coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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</li>

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

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> is <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a>: for every set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{e_{1},\dots ,e_{n}\}\subset E}">

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<annotation encoding="application/x-tex">{\displaystyle \{e_{1},\dots ,e_{n}\}\subset E}</annotation>

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</math><img src="./a68788a4be142acbb140689b6217e98ef4254bfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.817ex; height:2.843ex;" alt="{\displaystyle \{e_{1},\dots ,e_{n}\}\subset E}" loading="lazy"> of distinct elements, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{n}e_{n}=0_{M}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{n}e_{n}=0_{M}}</annotation>

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</math><img src="./15842adb884fdf283e37e51771b2605145375938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.515ex; height:2.509ex;" alt="{\displaystyle r_{1}e_{1}+r_{2}e_{2}+\cdots +r_{n}e_{n}=0_{M}}" loading="lazy"> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}=r_{2}=\cdots =r_{n}=0_{R}}">

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<annotation encoding="application/x-tex">{\displaystyle r_{1}=r_{2}=\cdots =r_{n}=0_{R}}</annotation>

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</math><img src="./3c1afc5137e230040ac1bd9d7fe26df310cd7110.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.232ex; height:2.509ex;" alt="{\displaystyle r_{1}=r_{2}=\cdots =r_{n}=0_{R}}" loading="lazy"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{M}}">

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

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</math><img src="./086b1974aa9acb66dcd7d9d876328f01081a58ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.122ex; height:2.509ex;" alt="{\displaystyle 0_{M}}" loading="lazy"> is the zero element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{R}}">

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

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</math><img src="./911229219f2ec31e016369e0ea56eef4b1f6b0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle 0_{R}}" loading="lazy"> is the zero element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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">).</li></ul>

A free module is a module with a basis.<a href="#cite_note-2">[2]</a>

An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M.

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

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

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</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"> has <a href="Invariant_basis_number" title="Invariant basis number">invariant basis number</a>, then by definition any two bases have the same cardinality. For example, nonzero commutative rings have invariant basis number. The cardinality of any (and therefore every) basis is called the rank of the free module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</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">. If this cardinality is finite, the free module is said to be free of finite rank, or free of rank n if the rank is known to be n.

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

Let R be a ring.

<ul><li>R is a free module of rank one over itself (either as a left or right module); any unit element is a basis.</li>

<li>More generally, If R is commutative, a nonzero ideal I of R is free if and only if it is a <a href="Principal_ideal" title="Principal ideal">principal ideal</a> generated by a <a href="Nonzerodivisor" class="mw-redirect" title="Nonzerodivisor">nonzerodivisor</a>, with a generator being a basis.<a href="#cite_note-3">[3]</a></li>

<li>Over a <a href="Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a> (e.g., <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">), a submodule of a free module is free.</li>

<li>If R is commutative, the polynomial ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X]}">

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

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</math><img src="./afeccd52deabb878398a8485755c3ceea80caf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle R[X]}" loading="lazy"> in indeterminate X is a free module with a possible basis 1, X, X2, ....</li>

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

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</math><img src="./13daf73f25e64310e7e5d67c5aa8dd4f97cb884f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle A[t]}" loading="lazy"> be a <a href="Polynomial_ring" title="Polynomial ring">polynomial ring</a> over a commutative ring A, f a <a href="Monic_polynomial" title="Monic polynomial">monic polynomial</a> of degree d there, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=A[t]/(f)}">

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<annotation encoding="application/x-tex">{\displaystyle B=A[t]/(f)}</annotation>

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</math><img src="./768151c0a8d11b44136db1811bee3ddd9ac55df5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.989ex; height:2.843ex;" alt="{\displaystyle B=A[t]/(f)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }">

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

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</math><img src="./e0b461aaf61091abd5d2c808931c48b8ff9647db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" loading="lazy"> the image of t in B. Then B contains A as a subring and is free as an A-module with a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,\xi ,\dots ,\xi ^{d-1}}">

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<annotation encoding="application/x-tex">{\displaystyle 1,\xi ,\dots ,\xi ^{d-1}}</annotation>

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</math><img src="./bf4cd6ef7fbd922de5a455ae8886e7de473da89d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.631ex; height:3.009ex;" alt="{\displaystyle 1,\xi ,\dots ,\xi ^{d-1}}" loading="lazy">.</li>

<li>For any non-negative integer n, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{n}=R\times \cdots \times R}">

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

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</math><img src="./c8b3d2cbfd56cfeaf4bd62b60aaff8d7fd920de4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.013ex; height:2.343ex;" alt="{\displaystyle R^{n}=R\times \cdots \times R}" loading="lazy">, the <a href="Direct_product#Direct_product_of_modules" title="Direct product">cartesian product</a> of n copies of R as a left R-module, is free. If R has <a href="Invariant_basis_number" title="Invariant basis number">invariant basis number</a>, then its <a href="Rank_of_a_module" class="mw-redirect" title="Rank of a module">rank</a> is n.</li>

<li>A <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the <a href="Baer%E2%80%93Specker_group" title="Baer–Specker group">Baer–Specker group</a>).</li>

<li>A finitely generated module over a commutative <a href="Local_ring" title="Local ring">local ring</a> is free if and only if it is <a href="Flat_module#Faithful_flatness" title="Flat module">faithfully flat</a>.<a href="#cite_note-4">[4]</a> Also, <a href="Kaplansky's_theorem_on_projective_modules" title="Kaplansky's theorem on projective modules">Kaplansky's theorem</a> states a projective module over a (possibly non-commutative) local ring is free.</li>

<li>Sometimes, whether a module is free or not is <a href="Undecidable_problem#Examples_of_undecidable_statements" title="Undecidable problem">undecidable</a> in the set-theoretic sense. A famous example is the <a href="Whitehead_problem" title="Whitehead problem">Whitehead problem</a>, which asks whether a Whitehead group is free or not. As it turns out, the problem is independent of ZFC.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Formal_linear_combinations">Formal linear combinations</h2></div>

Given a set E and ring R, there is a free R-module that has E as a basis: namely, the <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of copies of R indexed by E

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<annotation encoding="application/x-tex">{\displaystyle R^{(E)}=\bigoplus _{e\in E}R}</annotation>

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</math><img src="./3ac3de90ee538e14832ff65d9762fb202dee900a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.291ex; height:5.676ex;" alt="{\displaystyle R^{(E)}=\bigoplus _{e\in E}R}" loading="lazy">.</dd></dl>

Explicitly, it is the submodule of the <a href="Direct_product#Direct_product_of_modules" title="Direct product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{E}R}">

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<annotation encoding="application/x-tex">{\textstyle \prod _{E}R}</annotation>

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</math><img src="./4c08fc11c01eb79d2ae3c6e09c4898cb42756cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.833ex; height:3.009ex;" alt="{\textstyle \prod _{E}R}" loading="lazy"> (R is viewed as say a left module) that consists of the elements that have only finitely many nonzero components. One can <a href="Embedding" title="Embedding">embed</a> E into R(E) as a subset by identifying an element e with that of R(E) whose e-th component is 1 (the unity of R) and all the other components are zero. Then each element of R(E) can be written uniquely as

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<annotation encoding="application/x-tex">{\displaystyle \sum _{e\in E}c_{e}e,}</annotation>

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</math><img src="./e77dc2c0fca27c08ba811e80369def9276093882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:7.478ex; height:5.676ex;" alt="{\displaystyle \sum _{e\in E}c_{e}e,}" loading="lazy"></dd></dl>

where only finitely many <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{e}}">

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

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</math><img src="./6f4115c10cda4e7ee514243f54529571ff23ca66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.009ex;" alt="{\displaystyle c_{e}}" loading="lazy"> are nonzero. It is called a <a href="Formal_linear_combination" class="mw-redirect" title="Formal linear combination">formal linear combination</a> of elements of E.

A similar argument shows that every free left (resp. right) R-module is isomorphic to a direct sum of copies of R as left (resp. right) module.

<div class="mw-heading mw-heading3"><h3 id="Another_construction">Another construction</h3></div>

The free module R(E) may also be constructed in the following equivalent way.

Given a ring R and a set E, first as a set we let

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<mtext> for all but finitely many </mtext>

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<annotation encoding="application/x-tex">{\displaystyle R^{(E)}=\{f:E\to R\mid f(x)=0{\text{ for all but finitely many }}x\in E\}.}</annotation>

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</math><img src="./9efd697b50d643fa5c2227c530ca2ef8815214dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:62.476ex; height:3.343ex;" alt="{\displaystyle R^{(E)}=\{f:E\to R\mid f(x)=0{\text{ for all but finitely many }}x\in E\}.}" loading="lazy"></dd></dl>

We equip it with a structure of a left module such that the addition is defined by: for x in E,

</mstyle>

</mrow>

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

</semantics>

</math><img src="./cf80cb50218eac1e40d4a0908bd039db3bd0863c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.795ex; height:2.843ex;" alt="{\displaystyle (f+g)(x)=f(x)+g(x)}" loading="lazy"></dd></dl>

and the scalar multiplication by: for r in R and x in E,

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d21727ffb55476f406fdf73b601e8b75d20e2353.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.84ex; height:2.843ex;" alt="{\displaystyle (rf)(x)=rf(x)}" loading="lazy"></dd></dl>

Now, as an R-valued <a href="Function_(mathematics)" title="Function (mathematics)">function</a> on E, each f in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(E)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d5579f97648ac91d2d5c103542a6b0ef0730aa59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.531ex; height:2.843ex;" alt="{\displaystyle R^{(E)}}" loading="lazy"> can be written uniquely as

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=\sum _{e\in E}c_{e}\delta _{e}}</annotation>

</semantics>

</math><img src="./374ce6b12723bbb2ad12697b78fe835565a53101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.155ex; height:5.676ex;" alt="{\displaystyle f=\sum _{e\in E}c_{e}\delta _{e}}" loading="lazy"></dd></dl>

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

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6f4115c10cda4e7ee514243f54529571ff23ca66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.009ex;" alt="{\displaystyle c_{e}}" loading="lazy"> are in R and only finitely many of them are nonzero and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e18d0dbba5d6a01d4292950d6f49d12dcee61670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.031ex; height:2.676ex;" alt="{\displaystyle \delta _{e}}" loading="lazy"> is given as

<msub>

</mrow>

</msub>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{e}(x)={\begin{cases}1_{R}\quad {\mbox{if }}x=e\\0_{R}\quad {\mbox{if }}x\neq e\end{cases}}}</annotation>

</semantics>

</math><img src="./d57aaa33eadf2c2b88571210888e36f2c1e0d7eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.178ex; height:6.176ex;" alt="{\displaystyle \delta _{e}(x)={\begin{cases}1_{R}\quad {\mbox{if }}x=e\\0_{R}\quad {\mbox{if }}x\neq e\end{cases}}}" loading="lazy"></dd></dl>

(this is a variant of the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>). The above means that the subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\delta _{e}\mid e\in E\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\delta _{e}\mid e\in E\}}</annotation>

</semantics>

</math><img src="./3fb86bca3f39c13a3726cee2ba1c60b7aa8f2c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.993ex; height:2.843ex;" alt="{\displaystyle \{\delta _{e}\mid e\in E\}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(E)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d5579f97648ac91d2d5c103542a6b0ef0730aa59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.531ex; height:2.843ex;" alt="{\displaystyle R^{(E)}}" loading="lazy"> is a basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(E)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d5579f97648ac91d2d5c103542a6b0ef0730aa59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.531ex; height:2.843ex;" alt="{\displaystyle R^{(E)}}" loading="lazy">. The mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\mapsto \delta _{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./b9d7d00f8171106f9ccf65ff61f626c2fa9a16ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.728ex; height:2.676ex;" alt="{\displaystyle e\mapsto \delta _{e}}" loading="lazy"> is a <a href="Bijection" title="Bijection">bijection</a> between E and this basis. Through this bijection, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(E)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d5579f97648ac91d2d5c103542a6b0ef0730aa59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.531ex; height:2.843ex;" alt="{\displaystyle R^{(E)}}" loading="lazy"> is a free module with the basis E.

<div class="mw-heading mw-heading2"><h2 id="Universal_property">Universal property</h2></div>

The inclusion mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota :E\to R^{(E)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota :E\to R^{(E)}}</annotation>

</semantics>

</math><img src="./8b8d4c679b700284aca5669f9ba9f85b592477fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.681ex; height:2.843ex;" alt="{\displaystyle \iota :E\to R^{(E)}}" loading="lazy"> defined above is <a href="Universal_property" title="Universal property">universal</a> in the following sense. Given an arbitrary function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:E\to N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./dc52e613e943dff31f5ae106e8903b7cd349c949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.669ex; height:2.509ex;" alt="{\displaystyle f:E\to N}" loading="lazy"> from a set E to a left R-module N, there exists a unique <a href="Module_homomorphism" title="Module homomorphism">module homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}:R^{(E)}\to N}">

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {f}}:R^{(E)}\to N}</annotation>

</semantics>

</math><img src="./b054ebe8ab9e251095354d792342d3c621a9b948.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.63ex; height:3.343ex;" alt="{\displaystyle {\overline {f}}:R^{(E)}\to N}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\overline {f}}\circ \iota }">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ccb96645b0a4241c4330d54682041c799c2689b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.879ex; height:3.343ex;" alt="{\displaystyle f={\overline {f}}\circ \iota }" loading="lazy">; namely, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4de4554a5933ca51745edacd34a5a6aaea9a9f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.484ex; height:3.343ex;" alt="{\displaystyle {\overline {f}}}" loading="lazy"> is defined by the formula:

<mover>

</mover>

</mrow>

<mrow>

<mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\overline {f}}\left(\sum _{e\in E}r_{e}e\right)=\sum _{e\in E}r_{e}f(e)}</annotation>

</semantics>

</math><img src="./1ebe42d8e7c1a832488dd48b8cc3114fdfa53436.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.484ex; height:7.509ex;" alt="{\displaystyle {\overline {f}}\left(\sum _{e\in E}r_{e}e\right)=\sum _{e\in E}r_{e}f(e)}" loading="lazy"></dd></dl>

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

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4de4554a5933ca51745edacd34a5a6aaea9a9f4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.484ex; height:3.343ex;" alt="{\displaystyle {\overline {f}}}" loading="lazy"> is said to be obtained by extending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> by linearity. The uniqueness means that each R-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(E)}\to N}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{(E)}\to N}</annotation>

</semantics>

</math><img src="./447c9996de2bdf587d6754fcc27e172ce57ca8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.209ex; height:2.843ex;" alt="{\displaystyle R^{(E)}\to N}" loading="lazy"> is uniquely determined by its <a href="Restriction_(mathematics)" title="Restriction (mathematics)">restriction</a> to E.

As usual for universal properties, this defines R(E) <a href="Up_to" title="Up to">up to</a> a <a href="Canonical_isomorphism" class="mw-redirect" title="Canonical isomorphism">canonical isomorphism</a>. Also the formation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota :E\to R^{(E)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota :E\to R^{(E)}}</annotation>

</semantics>

</math><img src="./8b8d4c679b700284aca5669f9ba9f85b592477fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.681ex; height:2.843ex;" alt="{\displaystyle \iota :E\to R^{(E)}}" loading="lazy"> for each set E determines a <a href="Functor" title="Functor">functor</a>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{(-)}:{\textbf {Set}}\to R{\text{-}}{\mathsf {Mod}},\,E\mapsto R^{(E)}}</annotation>

</semantics>

</math><img src="./9776a7460caa3b16bfba962524e31901cb508bb3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.134ex; height:3.176ex;" alt="{\displaystyle R^{(-)}:{\textbf {Set}}\to R{\text{-}}{\mathsf {Mod}},\,E\mapsto R^{(E)}}" loading="lazy">,</dd></dl>

from the <a href="Category_of_sets" title="Category of sets">category of sets</a> to the category of left R-modules. It is called the <a href="Free_functor" class="mw-redirect" title="Free functor">free functor</a> and satisfies a natural relation: for each set E and a left module N,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{\textbf {Set}}(E,U(N))\simeq \operatorname {Hom} _{R}(R^{(E)},N),\,f\mapsto {\overline {f}}}</annotation>

</semantics>

</math><img src="./9e6a02378ac006104c4fe5d080f35a96c7e05a4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.655ex; height:3.509ex;" alt="{\displaystyle \operatorname {Hom} _{\textbf {Set}}(E,U(N))\simeq \operatorname {Hom} _{R}(R^{(E)},N),\,f\mapsto {\overline {f}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U:R{\text{-}}{\mathsf {Mod}}\to {\textbf {Set}}}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U:R{\text{-}}{\mathsf {Mod}}\to {\textbf {Set}}}</annotation>

</semantics>

</math><img src="./c47838b61fc2a6da58e356e0e136148f6e28bfe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.02ex; height:2.176ex;" alt="{\displaystyle U:R{\text{-}}{\mathsf {Mod}}\to {\textbf {Set}}}" loading="lazy"> is the <a href="Forgetful_functor" title="Forgetful functor">forgetful functor</a>, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(-)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./bffd9d2daae73dedee8d42abf63a9df4b9ddb8af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.554ex; height:2.843ex;" alt="{\displaystyle R^{(-)}}" loading="lazy"> is a <a href="Left_adjoint" class="mw-redirect" title="Left adjoint">left adjoint</a> of the forgetful functor.

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

Many statements true for free modules extend to certain larger classes of modules. <a href="Projective_module" title="Projective module">Projective modules</a> are direct summands of free modules. <a href="Flat_module" title="Flat module">Flat modules</a> are defined by the property that tensoring with them preserves exact sequences. <a href="Torsion-free_module" title="Torsion-free module">Torsion-free modules</a> form an even broader class. For a finitely generated module over a PID (such as Z), the properties free, projective, flat, and torsion-free are equivalent.

See <a href="Local_ring" title="Local ring">local ring</a>, <a href="Perfect_ring" title="Perfect ring">perfect ring</a> and <a href="Dedekind_ring" class="mw-redirect" title="Dedekind ring">Dedekind ring</a>.

<ul><li><a href="Free_object" title="Free object">Free object</a></li>

<li><a href="Free_presentation" title="Free presentation">free presentation</a></li>

<li><a href="Free_resolution" class="mw-redirect" title="Free resolution">free resolution</a></li>

<li><a href="Quillen%E2%80%93Suslin_theorem" title="Quillen–Suslin theorem">Quillen–Suslin theorem</a></li>

<li><a href="Stably_free_module" title="Stably free module">stably free module</a></li>

<li><a href="Generic_freeness" class="mw-redirect" title="Generic freeness">generic freeness</a></li></ul>

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

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

</style><cite id="CITEREFKeown1975" class="citation book cs1">Keown (1975). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=hC9iTw8DO7gC&pg=PA24&dq=%22Every+vector+space+is+free%22">An Introduction to Group Representation Theory</a>. p. 24.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHazewinkel1989" class="citation book cs1">Hazewinkel (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s9F71NJxwzoC&pg=PA110&dq=%22A+free+module+is+a+module+with+a+basis%22">Encyclopaedia of Mathematics, Volume 4</a>. p. 110.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Proof: Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> is free with a basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{j}|j\}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./e789af7325196ab4b3a4a52dc0018ad1c35a3022.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.169ex; height:3.009ex;" alt="{\displaystyle \{x_{j}|j\}}" loading="lazy">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\neq k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a08a8f7e7c65621ea80f6989770e39aa591d0886.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:5.294ex; height:2.676ex;" alt="{\displaystyle j\neq k}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}x_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./a111c1e93854a0ee2df054fc471d45d5b92e58b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.658ex; height:2.343ex;" alt="{\displaystyle x_{j}x_{k}}" loading="lazy"> must have the unique linear combination in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./5db47cb3d2f9496205a17a6856c91c1d3d363ccd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.239ex; height:2.343ex;" alt="{\displaystyle x_{j}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./6d2b88c64c76a03611549fb9b4cf4ed060b56002.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.418ex; height:2.009ex;" alt="{\displaystyle x_{k}}" loading="lazy">, which is not true. Thus, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\neq 0}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./4b987161191a84a74a32c91c33fbedf5be61979b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.433ex; height:2.676ex;" alt="{\displaystyle I\neq 0}" loading="lazy">, there is only one basis element which must be a nonzerodivisor. The converse is clear.<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \square }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./455831d58fa08f311b934d324adcff89a868b4e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \square }" loading="lazy">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFMatsumura1986">Matsumura 1986</a>, Theorem 7.10.

</li>

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

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

This article incorporates material from free vector space over a set on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.

<ul><li><cite id="CITEREFAdamson1972" class="citation book cs1">Adamson, Iain T. (1972). Elementary Rings and Modules. University Mathematical Texts. Oliver and Boyd. pp. 65–66. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-05-002192-3</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=0345993">0345993</a>.</cite></li>

<li><cite id="CITEREFKeown1975" class="citation book cs1">Keown, R. (1975). An Introduction to Group Representation Theory. Mathematics in science and engineering. Vol. 116. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-404250-6</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=0387387">0387387</a>.</cite></li>

<li><cite id="CITEREFGovorov2001" class="citation cs2">Govorov, V. E. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Free_module">"Free module"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>.</li>

<li><cite id="CITEREFMatsumura1986" class="citation book cs1"><a href="Hideyuki_Matsumura" title="Hideyuki Matsumura">Matsumura, Hideyuki</a> (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yJwNrABugDEC&pg=PA123">Commutative ring theory</a>. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-36764-6</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=0879273">0879273</a>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0603.13001">0603.13001</a>.</cite></li></ul>

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</style><div id="Dimension155" style="font-size:114%;margin:0 4em"><a href="Dimension" title="Dimension">Dimension</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensional spaces</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dimension_(vector_space)" title="Dimension (vector space)">Vector space</a></li>

<li><a href="Euclidean_space" title="Euclidean space">Euclidean space</a></li>

<li><a href="Affine_space" title="Affine space">Affine space</a></li>

<li><a href="Projective_space" title="Projective space">Projective space</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">Algebraic variety</a></li>

<li><a href="Spacetime" title="Spacetime">Spacetime</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other dimensions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Krull_dimension" title="Krull dimension">Krull</a></li>

<li><a href="Lebesgue_covering_dimension" title="Lebesgue covering dimension">Lebesgue covering</a></li>

<li><a href="Inductive_dimension" title="Inductive dimension">Inductive</a></li>

<li><a href="Hausdorff_dimension" title="Hausdorff dimension">Hausdorff</a></li>

<li><a href="Minkowski%E2%80%93Bouligand_dimension" title="Minkowski–Bouligand dimension">Minkowski</a></li>

<li><a href="Fractal_dimension" title="Fractal dimension">Fractal</a></li>

<li><a href="Degrees_of_freedom" title="Degrees of freedom">Degrees of freedom</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polytope" title="Polytope">Polytopes</a> and <a href="Shape" title="Shape">shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hyperplane" title="Hyperplane">Hyperplane</a></li>

<li><a href="Hypersurface" title="Hypersurface">Hypersurface</a></li>

<li><a href="Hypercube" title="Hypercube">Hypercube</a></li>

<li><a href="Hyperrectangle" title="Hyperrectangle">Hyperrectangle</a></li>

<li><a href="Demihypercube" title="Demihypercube">Demihypercube</a></li>

<li><a href="N-sphere" title="N-sphere">Hypersphere</a></li>

<li><a href="Cross-polytope" title="Cross-polytope">Cross-polytope</a></li>

<li><a href="Simplex" title="Simplex">Simplex</a></li>

<li><a href="Hyperpyramid" title="Hyperpyramid">Hyperpyramid</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Number systems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hypercomplex_number" title="Hypercomplex number">Hypercomplex numbers</a></li>

<li><a href="Cayley%E2%80%93Dickson_construction" title="Cayley–Dickson construction">Cayley–Dickson construction</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Dimensions by number</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Three-dimensional_space" title="Three-dimensional space">Three</a></li>

<li><a href="Seven-dimensional_space" title="Seven-dimensional space">Seven</a></li>

<li><a href="Eight-dimensional_space" title="Eight-dimensional space">Eight</a></li>

<li><a href="Dimension" title="Dimension">n-dimensions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hyperspace" title="Hyperspace">Hyperspace</a></li>

<li><a href="Codimension" title="Codimension">Codimension</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>Category</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-27" href="https://en.wikipedia.org/wiki/?title=Free_module&oldid=1302819735">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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