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Injective module

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In <a href="Mathematics" title="Mathematics">mathematics</a>, especially in the area of <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a> known as <a href="Module_theory" class="mw-redirect" title="Module theory">module theory</a>, an injective module is a <a href="Module_(mathematics)" title="Module (mathematics)">module</a> Q that shares certain desirable properties with the Z-module Q of all <a href="Rational_number" title="Rational number">rational numbers</a>. Specifically, if Q is a <a href="Submodule" class="mw-redirect" title="Submodule">submodule</a> of some other module, then it is already a <a href="Direct_summand" class="mw-redirect" title="Direct summand">direct summand</a> of that module; also, given a submodule of a module Y, any <a href="Module_homomorphism" title="Module homomorphism">module homomorphism</a> from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is <a href="Dual_(category_theory)" title="Dual (category theory)">dual</a> to that of <a href="Projective_module" title="Projective module">projective modules</a>. Injective modules were introduced in (<a href="#CITEREFBaer1940">Baer 1940</a>) and are discussed in some detail in the textbook (<a href="#CITEREFLam1999">Lam 1999</a>, §3).

Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: <a href="Injective_cogenerator" title="Injective cogenerator">Injective cogenerators</a> are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the <a href="#Injective_resolutions">injective dimension</a> and represent modules in the <a href="Derived_category" title="Derived category">derived category</a>. <a href="Injective_hull" title="Injective hull">Injective hulls</a> are maximal <a href="Essential_extension" title="Essential extension">essential extensions</a>, and turn out to be minimal injective extensions. Over a <a href="Noetherian_ring" title="Noetherian ring">Noetherian ring</a>, every injective module is uniquely a direct sum of <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a> modules, and their structure is well understood. An injective module over one ring may be not injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as <a href="Group_ring" title="Group ring">group rings</a> of <a href="Finite_group" title="Finite group">finite groups</a> over <a href="Field_(mathematics)" title="Field (mathematics)">fields</a>. Injective modules include <a href="Divisible_group" title="Divisible group">divisible groups</a> and are generalized by the notion of <a href="Injective_object" title="Injective object">injective objects</a> in <a href="Category_theory" title="Category theory">category theory</a>.

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

A left module Q over the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> R is injective if it satisfies one (and therefore all) of the following equivalent conditions:

<ul><li>If Q is a submodule of some other left R-module M, then there exists another submodule K of M such that M is the <a href="Direct_sum_of_modules" title="Direct sum of modules">internal direct sum</a> of Q and K, i.e. Q + K = M and Q ∩ K = {0}.</li>

<li>Any <a href="Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> 0 →Q → M → K → 0 of left R-modules <a href="Split_exact_sequence" title="Split exact sequence">splits</a>.</li>

<li>If X and Y are left R-modules, f : X → Y is an <a href="Injective" class="mw-redirect" title="Injective">injective</a> module homomorphism and g : X → Q is an arbitrary module homomorphism, then there exists a module homomorphism h : Y → Q such that hf = g, i.e. such that the following diagram <a href="Commutative_diagram" title="Commutative diagram">commutes</a>:</li></ul>

<ul><li>The <a href="Contravariant_functor" class="mw-redirect" title="Contravariant functor">contravariant</a> <a href="Hom_functor" title="Hom functor">Hom functor</a> Hom(-,Q) from the <a href="Category_theory" title="Category theory">category</a> of left R-modules to the category of <a href="Abelian_group" title="Abelian group">abelian groups</a> is <a href="Exact_functor" title="Exact functor">exact</a>.</li></ul>

Injective right R-modules are defined in complete analogy.

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

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

Trivially, the zero module {0} is injective.

Given a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> k, every k-<a href="Vector_space" title="Vector space">vector space</a> Q is an injective k-module. Reason: if Q is a subspace of V, we can find a <a href="Basis_of_a_vector_space" class="mw-redirect" title="Basis of a vector space">basis</a> of Q and extend it to a basis of V. The new extending basis vectors <a href="Linear_span" title="Linear span">span</a> a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map h in the above definition is typically not unique.

The rationals Q (with addition) form an injective abelian group (i.e. an injective Z-module). The <a href="Factor_group" class="mw-redirect" title="Factor group">factor group</a> Q/Z and the <a href="Circle_group" title="Circle group">circle group</a> are also injective Z-modules. The factor group Z/nZ for n > 1 is injective as a Z/nZ-module, but not injective as an abelian group.

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

More generally, for any <a href="Integral_domain" title="Integral domain">integral domain</a> R with field of fractions K, the R-module K is an injective R-module, and indeed the smallest injective R-module containing R. For any <a href="Dedekind_domain" title="Dedekind domain">Dedekind domain</a>, the <a href="Quotient_module" title="Quotient module">quotient module</a> K/R is also injective, and its <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a> summands are the <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localizations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}/R}">

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

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</math><img src="./e1a00baff6d4f7e0bcedce5ecfa4a71f80009f0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.745ex; height:3.009ex;" alt="{\displaystyle R_{\mathfrak {p}}/R}" loading="lazy"> for the nonzero <a href="Prime_ideal" title="Prime ideal">prime ideals</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">. The <a href="Zero_ideal" class="mw-redirect" title="Zero ideal">zero ideal</a> is also prime and corresponds to the injective K. In this way there is a 1-1 correspondence between prime ideals and indecomposable injective modules.

A particularly rich theory is available for <a href="Commutative_ring" title="Commutative ring">commutative</a> <a href="Noetherian_ring" title="Noetherian ring">noetherian rings</a> due to <a href="Eben_Matlis" title="Eben Matlis">Eben Matlis</a>, (<a href="#CITEREFLam1999">Lam 1999</a>, §3I). Every injective module is uniquely a direct sum of indecomposable injective modules, and the indecomposable injective modules are uniquely identified as the injective hulls of the quotients R/P where P varies over the <a href="Prime_spectrum" class="mw-redirect" title="Prime spectrum">prime spectrum</a> of the ring. The injective hull of R/P as an R-module is canonically an RP module, and is the RP-injective hull of R/P. In other words, it suffices to consider <a href="Local_ring" title="Local ring">local rings</a>. The <a href="Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of the injective hull of R/P is the <a href="Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">completion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {R}}_{P}}">

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

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</math><img src="./50d6ce07630260330b874fa0cabfaaa81c1a1c1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.23ex; height:3.176ex;" alt="{\displaystyle {\hat {R}}_{P}}" loading="lazy"> of R at P.<a href="#cite_note-1">[1]</a>

Two examples are the injective hull of the Z-module Z/pZ (the <a href="Pr%C3%BCfer_group" title="Prüfer group">Prüfer group</a>), and the injective hull of the k[x]-module k (the ring of inverse polynomials). The latter is easily described as k[x,x−1]/xk[x]. This module has a basis consisting of "inverse monomials", that is x−n for n = 0, 1, 2, …. Multiplication by scalars is as expected, and multiplication by x behaves normally except that x·1 = 0. The endomorphism ring is simply the ring of <a href="Formal_power_series" title="Formal power series">formal power series</a>.

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

If G is a <a href="Finite_group" title="Finite group">finite group</a> and k a field with <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0, then one shows in the theory of <a href="Group_representation" title="Group representation">group representations</a> that any subrepresentation of a given one is already a direct summand of the given one. Translated into module language, this means that all modules over the <a href="Group_ring" title="Group ring">group algebra</a> kG are injective. If the characteristic of k is not zero, the following example may help.

If A is a unital <a href="Associative_algebra" title="Associative algebra">associative algebra</a> over the field k with finite <a href="Dimension_of_a_vector_space" class="mw-redirect" title="Dimension of a vector space">dimension</a> over k, then Homk(−, k) is a <a href="Duality_of_categories" class="mw-redirect" title="Duality of categories">duality</a> between finitely generated left A-modules and finitely generated right A-modules. Therefore, the finitely generated injective left A-modules are precisely the modules of the form Homk(P, k) where P is a finitely generated projective right A-module. For <a href="Frobenius_algebra" title="Frobenius algebra">symmetric algebras</a>, the duality is particularly well-behaved and projective modules and injective modules coincide.

For any <a href="Artinian_ring" title="Artinian ring">Artinian ring</a>, just as for <a href="Commutative_ring" title="Commutative ring">commutative rings</a>, there is a 1-1 correspondence between prime ideals and indecomposable injective modules. The correspondence in this case is perhaps even simpler: a prime ideal is an annihilator of a unique simple module, and the corresponding indecomposable injective module is its <a href="Injective_hull" title="Injective hull">injective hull</a>. For finite-dimensional algebras over fields, these injective hulls are <a href="Finitely-generated_module" class="mw-redirect" title="Finitely-generated module">finitely-generated modules</a> (<a href="#CITEREFLam1999">Lam 1999</a>, §3G, §3J).

<div class="mw-heading mw-heading4"><h4 id="Computing_injective_hulls">Computing injective hulls</h4></div>

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"> is a Noetherian ring and <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"> is a prime ideal, set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=E(R/{\mathfrak {p}})}">

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

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</math><img src="./d367d100be8c139366784db14ecad5f9ab7e27ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.548ex; height:2.843ex;" alt="{\displaystyle E=E(R/{\mathfrak {p}})}" loading="lazy"> as the injective hull. The injective hull of <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="./e9da6e6f1e63ba7c169bec572ae2ca16f1c58936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {p}}}" loading="lazy"> over the Artinian ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {p}}^{k}}">

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

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</math><img src="./9c7ed5eebde17154224ce85bcb5596fc6d94d1df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.178ex; height:3.176ex;" alt="{\displaystyle R/{\mathfrak {p}}^{k}}" loading="lazy"> can be computed as the module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0:_{E}{\mathfrak {p}}^{k})}">

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<annotation encoding="application/x-tex">{\displaystyle (0:_{E}{\mathfrak {p}}^{k})}</annotation>

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</math><img src="./6bfa72c6fb66f50acd2d4d23f90eb96df2aa142d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.648ex; height:3.176ex;" alt="{\displaystyle (0:_{E}{\mathfrak {p}}^{k})}" loading="lazy">. It is a module of the same length as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {p}}^{k}}">

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

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</math><img src="./9c7ed5eebde17154224ce85bcb5596fc6d94d1df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.178ex; height:3.176ex;" alt="{\displaystyle R/{\mathfrak {p}}^{k}}" loading="lazy">.<a href="#cite_note-:0-2">[2]</a> In particular, for the standard graded ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\bullet }=k[x_{1},\ldots ,x_{n}]_{\bullet }}">

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<annotation encoding="application/x-tex">{\displaystyle R_{\bullet }=k[x_{1},\ldots ,x_{n}]_{\bullet }}</annotation>

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</math><img src="./e92f56eab6bdba0f6c8500f1ea5f48cb39956dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.586ex; height:2.843ex;" alt="{\displaystyle R_{\bullet }=k[x_{1},\ldots ,x_{n}]_{\bullet }}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}=(x_{1},\ldots ,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}=(x_{1},\ldots ,x_{n})}</annotation>

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</math><img src="./3ddee78e5831f5c101b493dabc05e43b122c8cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.181ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}=(x_{1},\ldots ,x_{n})}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\oplus _{i}{\text{Hom}}(R_{i},k)}">

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<annotation encoding="application/x-tex">{\displaystyle E=\oplus _{i}{\text{Hom}}(R_{i},k)}</annotation>

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</math><img src="./3047a9454afcc60cb830079ed475a6b0b97d06fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.941ex; height:2.843ex;" alt="{\displaystyle E=\oplus _{i}{\text{Hom}}(R_{i},k)}" loading="lazy"> is an injective module, giving the tools for computing the indecomposable injective modules for artinian rings over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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<div class="mw-heading mw-heading4"><h4 id="Self-injectivity">Self-injectivity</h4></div>

An Artin local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,{\mathfrak {m}},K)}">

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

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</math><img src="./1fad8724a1e8907081e6d5dbccdc30deb73c6385.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.49ex; height:2.843ex;" alt="{\displaystyle (R,{\mathfrak {m}},K)}" loading="lazy"> is injective over itself if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle soc(R)}">

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

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</math><img src="./c52d11d253a1aee7c553632b79d0c6385b4268a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.798ex; height:2.843ex;" alt="{\displaystyle soc(R)}" loading="lazy"> is a 1-dimensional vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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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">. This implies every local Gorenstein ring which is also Artin is injective over itself since has a 1-dimensional socle.<a href="#cite_note-3">[3]</a> A simple non-example is the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\mathbb {C} [x,y]/(x^{2},xy,y^{2})}">

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

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</math><img src="./d2e787254550e894db6de03a7a0a9c82b386c6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.477ex; height:3.176ex;" alt="{\displaystyle R=\mathbb {C} [x,y]/(x^{2},xy,y^{2})}" loading="lazy"> which has maximal ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)}">

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

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</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"> and residue field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

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

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</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">. Its socle is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \cdot x\oplus \mathbb {C} \cdot y}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \cdot x\oplus \mathbb {C} \cdot y}</annotation>

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</math><img src="./c1fc65d92191c0732ac6fb5d3b825fad9da7c4c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.04ex; height:2.509ex;" alt="{\displaystyle \mathbb {C} \cdot x\oplus \mathbb {C} \cdot y}" loading="lazy">, which is 2-dimensional. The residue field has the injective hull <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Hom}}_{\mathbb {C} }(\mathbb {C} \cdot x\oplus \mathbb {C} \cdot y,\mathbb {C} )}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{Hom}}_{\mathbb {C} }(\mathbb {C} \cdot x\oplus \mathbb {C} \cdot y,\mathbb {C} )}</annotation>

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</math><img src="./0dbf5fc891aab72784a0ac711a597a06d54f2749.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.821ex; height:2.843ex;" alt="{\displaystyle {\text{Hom}}_{\mathbb {C} }(\mathbb {C} \cdot x\oplus \mathbb {C} \cdot y,\mathbb {C} )}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Modules_over_Lie_algebras">Modules over Lie algebras</h3></div>

For a Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> of characteristic 0, the category of modules <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}({\mathfrak {g}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d74d7d45a0c52c84797b4bbf6b6daf1fb5dbc51e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.772ex; height:2.843ex;" alt="{\displaystyle {\mathcal {M}}({\mathfrak {g}})}" loading="lazy"> has a relatively straightforward description of its injective modules.<a href="#cite_note-4">[4]</a> Using the universal enveloping algebra any injective <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">-module can be constructed from the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">-module<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}</annotation>

</semantics>

</math><img src="./4f54248251132d47a2b44b5c2b418f459ba04fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.324ex; height:2.843ex;" alt="{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}" loading="lazy"></blockquote>for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Note this vector space has a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">-module structure from the injection<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}\hookrightarrow U({\mathfrak {g}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./d4fce843c23071b4db8e622f147f81fbc7096638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.842ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {g}}\hookrightarrow U({\mathfrak {g}})}" loading="lazy"></blockquote>In fact, every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">-module has an injection into some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}</annotation>

</semantics>

</math><img src="./4f54248251132d47a2b44b5c2b418f459ba04fea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.324ex; height:2.843ex;" alt="{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}" loading="lazy"> and every injective <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy">-module is a direct summand of some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Hom}}_{k}(U({\mathfrak {g}}),V)}</annotation>

</semantics>

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

<div class="mw-heading mw-heading3"><h3 id="Structure_theorem_for_commutative_Noetherian_rings">Structure theorem for commutative Noetherian rings</h3></div>

Over a commutative <a href="Noetherian_ring" title="Noetherian ring">Noetherian ring</a> <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">, every injective module is a direct sum of indecomposable injective modules and every indecomposable injective module is the injective hull of the residue field at a prime <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">. That is, for an injective <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\in {\text{Mod}}(R)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\in {\text{Mod}}(R)}</annotation>

</semantics>

</math><img src="./018b6f1aa87c733eb1fc46b9f9a23bc2632731e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.172ex; height:2.843ex;" alt="{\displaystyle I\in {\text{Mod}}(R)}" loading="lazy"> , there is an isomorphism<blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\cong \bigoplus _{i}E(R/{\mathfrak {p}}_{i})}">

</mrow>

</munder>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I\cong \bigoplus _{i}E(R/{\mathfrak {p}}_{i})}</annotation>

</semantics>

</math><img src="./9a0ad76be78b13f2bd5d2009420f5b95c8c49d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.641ex; height:5.509ex;" alt="{\displaystyle I\cong \bigoplus _{i}E(R/{\mathfrak {p}}_{i})}" loading="lazy"></blockquote>where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(R/{\mathfrak {p}}_{i})}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(R/{\mathfrak {p}}_{i})}</annotation>

</semantics>

</math><img src="./b198cec45ad98213fb7d44aafe85021658c473fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.474ex; height:2.843ex;" alt="{\displaystyle E(R/{\mathfrak {p}}_{i})}" loading="lazy"> are the injective hulls of the modules <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {p}}_{i}}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./43c54621ccc8a1cc693e5a145c564b32551287b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.889ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {p}}_{i}}" loading="lazy">.<a href="#cite_note-5">[5]</a> In addition, if <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 the injective hull of some 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"> then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./ebc456e8ce59c0b5f16401170ef68c996c25d512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.962ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{i}}" loading="lazy"> are the associated primes of <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">.<a href="#cite_note-:0-2">[2]</a>

<div class="mw-heading mw-heading3"><h3 id="Submodules,_quotients,_products,_and_sums,_Bass-Papp_Theorem">Submodules, quotients, products, and sums, Bass-Papp Theorem</h3></div>

Any <a href="Product_(category_theory)" title="Product (category theory)">product</a> of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective (<a href="#CITEREFLam1999">Lam 1999</a>, p. 61). Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sums</a> of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is <a href="Artinian_ring" title="Artinian ring">Artinian</a> <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple</a> (<a href="#CITEREFGolanHead1991">Golan & Head 1991</a>, p. 152); every factor module of every injective module is injective if and only if the ring is <a href="Hereditary_ring" title="Hereditary ring">hereditary</a>, (<a href="#CITEREFLam1999">Lam 1999</a>, Th. 3.22).

Bass-Papp Theorem states that every infinite direct sum of right (left) injective modules is injective if and only if the ring is right (left) <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a>, (<a href="#CITEREFLam1999">Lam 1999</a>, p. 80-81, Th 3.46).<a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Baer's_criterion">Baer's criterion</h3></div>

In Baer's original paper, he proved a useful result, usually known as Baer's Criterion, for checking whether a module is injective: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">left ideal</a> I of R can be extended to all of R.

Using this criterion, one can show that Q is an injective <a href="Abelian_group" title="Abelian group">abelian group</a> (i.e. an injective module over Z). More generally, an abelian group is injective if and only if it is <a href="Divisible_module" class="mw-redirect" title="Divisible module">divisible</a>. More generally still: a module over a <a href="Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a> is injective if and only if it is divisible (the case of vector spaces is an example of this theorem, as every field is a principal ideal domain and every vector space is divisible). Over a general integral domain, we still have one implication: every injective module over an integral domain is divisible.

Baer's criterion has been refined in many ways (<a href="#CITEREFGolanHead1991">Golan & Head 1991</a>, p. 119), including a result of (<a href="#CITEREFSmith1981">Smith 1981</a>) and (<a href="#CITEREFVámos1983">Vámos 1983</a>) that for a commutative Noetherian ring, it suffices to consider only <a href="Prime_ideal" title="Prime ideal">prime ideals</a> I. The dual of Baer's criterion, which would give a test for projectivity, is false in general. For instance, the Z-module Q satisfies the dual of Baer's criterion but is not projective.

<div class="mw-heading mw-heading3"><h3 id="Injective_cogenerators">Injective cogenerators</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Injective_cogenerator" title="Injective cogenerator">injective cogenerator</a></div>

Maybe the most important injective module is the abelian group Q/Z. It is an <a href="Injective_cogenerator" title="Injective cogenerator">injective cogenerator</a> in the <a href="Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a>, which means that it is injective and any other module is contained in a suitably large product of copies of Q/Z. So in particular, every abelian group is a subgroup of an injective one. It is quite significant that this is also true over any ring: every module is a submodule of an injective one, or "the category of left R-modules has enough injectives." To prove this, one uses the peculiar properties of the abelian group Q/Z to construct an injective cogenerator in the category of left R-modules.

For a left R-module M, the so-called "character module" M+ = HomZ(M,Q/Z) is a right R-module that exhibits an interesting duality, not between injective modules and <a href="Projective_module" title="Projective module">projective modules</a>, but between injective modules and <a href="Flat_module" title="Flat module">flat modules</a> (<a href="#CITEREFEnochsJenda2000">Enochs & Jenda 2000</a>, pp. 78–80). For any ring R, a left R-module is flat if and only if its character module is injective. If R is left noetherian, then a left R-module is injective if and only if its character module is flat.

<div class="mw-heading mw-heading3"><h3 id="Injective_hulls">Injective hulls</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Injective_hull" title="Injective hull">injective hull</a></div>

The <a href="Injective_hull" title="Injective hull">injective hull</a> of a module is the smallest injective module containing the given one and was described in (<a href="#CITEREFEckmannSchopf1953">Eckmann & Schopf 1953</a>).

One can use injective hulls to define a minimal injective resolution (see below). If each term of the injective resolution is the injective hull of the cokernel of the previous map, then the injective resolution has minimal length.

<div class="mw-heading mw-heading3"><h3 id="Injective_resolutions">Injective resolutions</h3></div>

Every module M also has an injective <a href="Resolution_(algebra)" title="Resolution (algebra)">resolution</a>: an <a href="Exact_sequence" title="Exact sequence">exact sequence</a> of the form

where the I j are injective modules. Injective resolutions can be used to define <a href="Derived_functor" title="Derived functor">derived functors</a> such as the <a href="Ext_functor" title="Ext functor">Ext functor</a>.

The length of a finite injective resolution is the first index n such that In is nonzero and Ii = 0 for i greater than n. If a module M admits a finite injective resolution, the minimal length among all finite injective resolutions of M is called its injective dimension and denoted id(M). If M does not admit a finite injective resolution, then by convention the injective dimension is said to be infinite. (<a href="#CITEREFLam1999">Lam 1999</a>, §5C) As an example, consider a module M such that id(M) = 0. In this situation, the exactness of the sequence 0 → M → I0 → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is injective.<a href="#cite_note-7">[7]</a>

Equivalently, the injective dimension of M is the minimal integer (if there is such, otherwise ∞) n such that ExtN A(–,M) = 0 for all N > n.

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

Every injective submodule of an injective module is a direct summand, so it is important to understand <a href="Indecomposable_module" title="Indecomposable module">indecomposable</a> injective modules, (<a href="#CITEREFLam1999">Lam 1999</a>, §3F).

Every indecomposable injective module has a <a href="Local_ring" title="Local ring">local</a> <a href="Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a>. A module is called a <a href="Uniform_module" title="Uniform module">uniform module</a> if every two nonzero submodules have nonzero intersection. For an injective module M the following are equivalent:

<ul><li>M is indecomposable</li>

<li>M is nonzero and is the injective hull of every nonzero submodule</li>

<li>M is uniform</li>

<li>M is the injective hull of a uniform module</li>

<li>M is the injective hull of a uniform <a href="Cyclic_module" title="Cyclic module">cyclic module</a></li>

<li>M has a local endomorphism ring</li></ul>

Over a Noetherian ring, every injective module is the direct sum of (uniquely determined) indecomposable injective modules. Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in (<a href="#CITEREFMatlis1958">Matlis 1958</a>). The indecomposable injective modules are the injective hulls of the modules R/p for p a prime ideal of the ring R. Moreover, the injective hull M of R/p has an increasing filtration by modules Mn given by the annihilators of the ideals pn, and Mn+1/Mn is isomorphic as finite-dimensional vector space over the quotient field k(p) of R/p to HomR/p(pn/pn+1, k(p)).

<div class="mw-heading mw-heading3"><h3 id="Change_of_rings">Change of rings</h3></div>

It is important to be able to consider modules over <a href="Subring" title="Subring">subrings</a> or <a href="Quotient_ring" title="Quotient ring">quotient rings</a>, especially for instance <a href="Polynomial_ring" title="Polynomial ring">polynomial rings</a>. In general, this is difficult, but a number of results are known, (<a href="#CITEREFLam1999">Lam 1999</a>, p. 62).

Let S and R be rings, and P be a left-R, right-S <a href="Bimodule" title="Bimodule">bimodule</a> that is <a href="Flat_module" title="Flat module">flat</a> as a left-R module. For any injective right S-module M, the set of <a href="Module_homomorphism" title="Module homomorphism">module homomorphisms</a> HomS( P, M ) is an injective right R-module. The same statement holds of course after interchanging left- and right- attributes.

For instance, if R is a subring of S such that S is a flat R-module, then every injective S-module is an injective R-module. In particular, if R is an integral domain and S its <a href="Field_of_fractions" title="Field of fractions">field of fractions</a>, then every vector space over S is an injective R-module. Similarly, every injective R[x]-module is an injective R-module.

In the opposite direction, a ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./db23d001e8cdb67ac9b7e2d97f0de85c1e3f9cdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle f:S\to R}" loading="lazy"> makes R into a left-R, right-S bimodule, by left and right multiplication. Being <a href="Free_module" title="Free module">free</a> over itself R is also <a href="Flat_module#Free_and_projective_modules" title="Flat module">flat</a> as a left R-module. Specializing the above statement for P = R, it says that when M is an injective right S-module the <a href="Coinduced_module" class="mw-redirect" title="Coinduced module">coinduced module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{*}M=\mathrm {Hom} _{S}(R,M)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{*}M=\mathrm {Hom} _{S}(R,M)}</annotation>

</semantics>

</math><img src="./be612542ec9105c1d620fe4e283bd09935975f3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.917ex; height:2.843ex;" alt="{\displaystyle f_{*}M=\mathrm {Hom} _{S}(R,M)}" loading="lazy"> is an injective right R-module. Thus, coinduction over f produces injective R-modules from injective S-modules.

For quotient rings R/I, the change of rings is also very clear. An R-module is an R/I-module precisely when it is annihilated by I. The submodule annI(M) = { m in M : im = 0 for all i in I } is a left submodule of the left R-module M, and is the largest submodule of M that is an R/I-module. If M is an injective left R-module, then annI(M) is an injective left R/I-module. Applying this to R=Z, I=nZ and M=Q/Z, one gets the familiar fact that Z/nZ is injective as a module over itself. While it is easy to convert injective R-modules into injective R/I-modules, this process does not convert injective R-resolutions into injective R/I-resolutions, and the homology of the resulting complex is one of the early and fundamental areas of study of relative homological algebra.

The textbook (<a href="#CITEREFRotman1979">Rotman 1979</a>, p. 103) has an erroneous proof that <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization</a> preserves injectives, but a counterexample was given in (<a href="#CITEREFDade1981">Dade 1981</a>).

<div class="mw-heading mw-heading3"><h3 id="Self-injective_rings">Self-injective rings</h3></div>

Every ring with unity is a <a href="Free_module" title="Free module">free module</a> and hence is a <a href="Projective_module" title="Projective module">projective</a> as a module over itself, but it is rarer for a ring to be injective as a module over itself, (<a href="#CITEREFLam1999">Lam 1999</a>, §3B). If a ring is injective over itself as a right module, then it is called a right self-injective ring. Every <a href="Frobenius_algebra" title="Frobenius algebra">Frobenius algebra</a> is self-injective, but no <a href="Integral_domain" title="Integral domain">integral domain</a> that is not a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> is self-injective. Every proper <a href="Quotient_ring" title="Quotient ring">quotient</a> of a <a href="Dedekind_domain" title="Dedekind domain">Dedekind domain</a> is self-injective.

A right <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a>, right self-injective ring is called a <a href="Quasi-Frobenius_ring" title="Quasi-Frobenius ring">quasi-Frobenius ring</a>, and is two-sided <a href="Artinian_ring" title="Artinian ring">Artinian</a> and two-sided injective, (<a href="#CITEREFLam1999">Lam 1999</a>, Th. 15.1). An important module theoretic property of quasi-Frobenius rings is that the projective modules are exactly the injective modules.

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

<div class="mw-heading mw-heading3"><h3 id="Injective_objects">Injective objects</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Injective_object" title="Injective object">injective object</a></div>

One also talks about <a href="Injective_object" title="Injective object">injective objects</a> in <a href="Category_(mathematics)" title="Category (mathematics)">categories</a> more general than module categories, for instance in <a href="Functor_category" title="Functor category">functor categories</a> or in categories of <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> of OX-modules over some <a href="Ringed_space" title="Ringed space">ringed space</a> (X,OX). The following general definition is used: an object Q of the category C is injective if for any <a href="Monomorphism" title="Monomorphism">monomorphism</a> f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g.

<div class="mw-heading mw-heading3"><h3 id="Divisible_groups">Divisible groups</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Divisible_group" title="Divisible group">divisible group</a></div>

The notion of injective object in the category of abelian groups was studied somewhat independently of injective modules under the term <a href="Divisible_group" title="Divisible group">divisible group</a>. Here a Z-module M is injective if and only if n⋅M = M for every nonzero integer n. Here the relationships between <a href="Flat_module" title="Flat module">flat modules</a>, <a href="Pure_submodule" title="Pure submodule">pure submodules</a>, and injective modules is more clear, as it simply refers to certain divisibility properties of module elements by integers.

<div class="mw-heading mw-heading3"><h3 id="Pure_injectives">Pure injectives</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Pure_injective_module" class="mw-redirect" title="Pure injective module">pure injective module</a></div>

In relative homological algebra, the extension property of homomorphisms may be required only for certain submodules, rather than for all. For instance, a <a href="Pure_injective_module" class="mw-redirect" title="Pure injective module">pure injective module</a> is a module in which a homomorphism from a <a href="Pure_submodule" title="Pure submodule">pure submodule</a> can be extended to the whole module.

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

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/08Z6">"Lemma 47.7.5 (08Z6)—The Stacks project"</a>. stacks.math.columbia.edu. Retrieved 2020-02-25.</cite>

</li>

<li id="cite_note-:0-2">^ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> <cite id="CITEREFEisenbud" class="citation book cs1">Eisenbud. Introduction to Commutative Algebra. pp. 624, 625.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.math.purdue.edu/~walther/snowbird/inj.pdf">"Injective Modules"</a> (PDF). p. 10.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFVogan" class="citation web cs1">Vogan, David. <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~dav/cohom.pdf">"Lie Algebra Cohomology"</a> (PDF).</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/08YA">"Structure of injective modules over Noetherian rings"</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> This is the <a href="Hyman_Bass" title="Hyman Bass">Bass</a>-Papp theorem, see (<a href="#CITEREFPapp1959">Papp 1959</a>) and (<a href="#CITEREFChase1960">Chase 1960</a>)

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> A module isomorphic to an injective module is of course injective.

</li>

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

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

<ul><li><cite id="CITEREFAndersonFuller1992" class="citation cs2">Anderson, Frank Wylie; Fuller, Kent R (1992), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PswhrD_wUIkC">Rings and Categories of Modules</a>, 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>978-0-387-97845-1</bdi>, retrieved 30 July 2016</cite></li>

<li><cite id="CITEREFEnochsJenda2000" class="citation cs2">Enochs, Edgar E.; <a href="Overtoun_Jenda" title="Overtoun Jenda">Jenda, Overtoun M. G.</a> (2000), Relative homological algebra, de Gruyter Expositions in Mathematics, vol. 30, Berlin: Walter de Gruyter & Co., <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110803662">10.1515/9783110803662</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-11-016633-0</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=1753146">1753146</a></cite></li>

<li><cite id="CITEREFGolanHead1991" class="citation cs2">Golan, Jonathan S.; Head, Tom (1991), <a rel="nofollow" class="external text" href="https://archive.org/details/modulesstructure0000gola">Modules and the structure of rings</a>, Monographs and Textbooks in Pure and Applied Mathematics, vol. 147, Marcel Dekker, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-8555-0</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=1201818">1201818</a></cite></li>

<li><cite id="CITEREFLam1999" class="citation cs2">Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0525-8">10.1007/978-1-4612-0525-8</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98428-5</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=1653294">1653294</a></cite></li>

<li><cite id="CITEREFRotman1979" class="citation cs2">Rotman, Joseph J. (1979), An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Boston, MA: <a href="Academic_Press" title="Academic Press">Academic Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-599250-3</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=0538169">0538169</a></cite></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Primary_sources">Primary sources</h3></div>

<ul><li><cite id="CITEREFBaer1940" class="citation cs2"><a href="Reinhold_Baer" title="Reinhold Baer">Baer, Reinhold</a> (1940), "Abelian groups that are direct summands of every containing abelian group", <a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a>, 46 (10): 800–807, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1940-07306-9">10.1090/S0002-9904-1940-07306-9</a>, <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=0002886">0002886</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:0024.14902">0024.14902</a></cite></li>

<li><cite id="CITEREFChase1960" class="citation cs2">Chase, Stephen U. (1960), "Direct products of modules", <a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 97 (3), American Mathematical Society, Vol. 97, No. 3: 457–473, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1993382">10.2307/1993382</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/1993382">1993382</a>, <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=0120260">0120260</a></cite></li>

<li><cite id="CITEREFDade1981" class="citation cs2"><a href="Everett_C._Dade" title="Everett C. Dade">Dade, Everett C.</a> (1981), "Localization of injective modules", <a href="Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a>, 69 (2): 416–425, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-8693%2881%2990213-1">10.1016/0021-8693(81)90213-1</a>, <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=0617087">0617087</a></cite></li>

<li><cite id="CITEREFEckmannSchopf1953" class="citation cs2"><a href="Beno_Eckmann" title="Beno Eckmann">Eckmann, B.</a>; Schopf, A. (1953), "Über injektive Moduln", <a href="Archiv_der_Mathematik" title="Archiv der Mathematik">Archiv der Mathematik</a>, 4 (2): 75–78, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01899665">10.1007/BF01899665</a>, <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=0055978">0055978</a></cite></li>

<li><cite id="CITEREFLambek1963" class="citation cs2"><a href="Joachim_Lambek" title="Joachim Lambek">Lambek, Joachim</a> (1963), <a rel="nofollow" class="external text" href="http://www.cms.math.ca/cjm/v15/p363">"On Utumi's ring of quotients"</a>, <a href="Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a>, 15: 363–370, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1963-041-4">10.4153/CJM-1963-041-4</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0008-414X">0008-414X</a>, <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=0147509">0147509</a></cite></li>

<li><cite id="CITEREFMatlis1958" class="citation cs2"><a href="Eben_Matlis" title="Eben Matlis">Matlis, Eben</a> (1958), "Injective modules over Noetherian rings", <a href="Pacific_Journal_of_Mathematics" title="Pacific Journal of Mathematics">Pacific Journal of Mathematics</a>, 8: 511–528, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1958.8.511">10.2140/pjm.1958.8.511</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0030-8730">0030-8730</a>, <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=0099360">0099360</a></cite></li>

<li><cite id="CITEREFOsofsky1964" class="citation cs2"><a href="Barbara_L._Osofsky" title="Barbara L. Osofsky">Osofsky, B. L.</a> (1964), "On ring properties of injective hulls", <a href="Canadian_Mathematical_Bulletin" title="Canadian Mathematical Bulletin">Canadian Mathematical Bulletin</a>, 7: 405–413, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCMB-1964-039-3">10.4153/CMB-1964-039-3</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0008-4395">0008-4395</a>, <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=0166227">0166227</a></cite></li>

<li><cite id="CITEREFPapp1959" class="citation cs2">Papp, Zoltán (1959), "On algebraically closed modules", <a href="Publicationes_Mathematicae_Debrecen" title="Publicationes Mathematicae Debrecen">Publicationes Mathematicae Debrecen</a>, 6: 311–327, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0033-3883">0033-3883</a>, <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=0121390">0121390</a></cite></li>

<li><cite id="CITEREFSmith1981" class="citation cs2">Smith, P. F. (1981), "Injective modules and prime ideals", Communications in Algebra, 9 (9): 989–999, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00927878108822627">10.1080/00927878108822627</a>, <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=0614468">0614468</a></cite></li>

<li><cite id="CITEREFUtumi1956" class="citation cs2">Utumi, Yuzo (1956), "On quotient rings", Osaka Journal of Mathematics, 8: 1–18, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0030-6126">0030-6126</a>, <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=0078966">0078966</a></cite></li>

<li><cite id="CITEREFVámos1983" class="citation cs2">Vámos, P. (1983), "Ideals and modules testing injectivity", Communications in Algebra, 11 (22): 2495–2505, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00927878308822975">10.1080/00927878308822975</a>, <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=0733337">0733337</a></cite></li></ul>

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