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Injective module

In homological algebra, an injective module over a ring R is an R-module I such that the contravariant Hom functor \operatorname{Hom}_R(-, I) from the category of R-modules to abelian groups is exact, meaning that for any injective homomorphism M \to N of R-modules, the induced map \operatorname{Hom}_R(N, I) \to \operatorname{Hom}_R(M, I) is surjective.^[1] This property ensures that maps into I can always be extended across inclusions, providing a dual notion to projective modules where maps out of them lift through surjections.^[2] A key characterization of injective modules over commutative rings is Baer's criterion, which states that an R-module I is injective if and only if, for every ideal \mathfrak{a} \subseteq R and every R-module homomorphism f: \mathfrak{a} \to I, there exists an extension \tilde{f}: R \to I such that \tilde{f}|_{\mathfrak{a}} = f.^[1] This criterion simplifies verification of injectivity by reducing it to extensions over ideals. Additionally, products of injective modules are injective, mirroring how direct sums of projectives behave, and every ring R has enough injective modules, meaning any R-module embeds into an injective one.^[3]^[1] Injective modules play a central role in constructing injective resolutions, which are used to compute derived functors like \operatorname{Ext} groups; specifically, I is injective if and only if \operatorname{Ext}^1_R(M, I) = 0 for all R-modules M.^[1] Over commutative Noetherian rings, every injective module decomposes as a direct sum of indecomposable injectives of the form E_R(R/\mathfrak{p}) for prime ideals \mathfrak{p}, where E_R(-) denotes the injective hull.^[2] Classic examples include the rationals \mathbb{Q} and the Prüfer p-group \mathbb{Z}(p^\infty) as injective \mathbb{Z}-modules, which are precisely the divisible abelian groups.^[3]

Definition and Characterizations

Formal Definition

In the category of left modules over an associative ring R with identity, an injective module is defined via an extension property that dualizes the lifting property of projective modules. Specifically, a left R-module E is injective if, for every left ideal I \subseteq R and every R-module homomorphism f: I \to E, there exists an R-module homomorphism \tilde{f}: R \to E such that \tilde{f}|_I = f. This condition ensures that maps into E from submodules of R can always be extended to the whole ring, reflecting the module's "universal" receptivity for homomorphisms. This definition is equivalent to the functor \operatorname{Hom}_R(-, E) being exact: for any short exact sequence $0 \to A \to B \to C \to 0 of left R-modules, the induced sequence $0 \to \operatorname{Hom}_R(C, E) \to \operatorname{Hom}_R(B, E) \to \operatorname{Hom}_R(A, E) \to 0 is also exact.^[1] The exactness captures how E preserves limits in the category, turning injections into surjections in the Hom functor. The concept of injective modules was introduced by Reinhold Baer in 1940, initially in the context of abelian groups as those that are direct summands of every containing abelian group, serving as a categorical dual to projective modules which lift homomorphisms over surjections.^[4] This duality underscores injectives' role in homological algebra, where they facilitate resolutions and computations of Ext functors, assuming familiarity with basic module theory and homomorphisms.

Baer's Criterion

Baer's criterion, named after Reinhold Baer who introduced the concept in the context of abelian groups, provides a computable characterization of injective modules over an arbitrary associative ring R with identity. Specifically, a left R-module E is injective if and only if for every left ideal I of R and every R-module homomorphism f: I \to E, there exists an R-module homomorphism \tilde{f}: R \to E such that the restriction of \tilde{f} to I equals f.^[2] This condition leverages the fact that ideals are particular submodules of the free module R, simplifying the test for injectivity compared to verifying the extension property for arbitrary submodules. The sufficiency of the criterion (that extension from ideals implies full injectivity) is proved using Zorn's lemma applied to the partially ordered set of pairs (M', g), where M \subseteq M' \subseteq N for submodules M \subseteq N and g: M' \to E is an extension of the given f: M \to E. A maximal such pair exists by Zorn's lemma; if M' \neq N, select x \in N \setminus M' and form the left ideal \mathrm{Ann}_R(N/M')(x) = \{ r \in R \mid r x \in M' \}. Define h: \mathrm{Ann}_R(N/M')(x) \to E by h(r) = g(r x). By the hypothesis, extend h to \tilde{h}: R \to E. Then define a new extension on M'' = M' + R x by g'(y + s x) = g(y) + \tilde{h}(s) for y \in M', s \in R; well-definedness follows because if s x \in M', then s \in \mathrm{Ann}_R(N/M')(x), so \tilde{h}(s) = h(s) = g(s x), contradicting maximality. The necessity direction is immediate, as ideals are submodules of R.^[2]^[5] For commutative rings R, the criterion admits a useful refinement: it suffices to verify the extension property for finitely generated ideals. This follows because any homomorphism from an arbitrary ideal I to E can be constructed inductively from maps on finite subsets of generators, using the commutativity to ensure compatibility of extensions across the generators of I, and the property holds for direct limits of such extensions.^[2] A concrete illustration arises in the category of \mathbb{Z}-modules (abelian groups), where the ideals of \mathbb{Z} are the principal ideals n\mathbb{Z} for n \geq 0. To confirm that \mathbb{Q} is injective, consider a homomorphism f: n\mathbb{Z} \to \mathbb{Q} for n > 0; without loss of generality, f(n) = q for some q \in \mathbb{Q}, extended \mathbb{Z}-linearly by f(n k) = k q. Define \tilde{f}: \mathbb{Z} \to \mathbb{Q} by \tilde{f}(m) = (m/n) q; then \tilde{f}(n k) = k q = f(n k), verifying the extension. For n=0, the zero map extends trivially. This computation demonstrates \mathbb{Q}'s injectivity directly via the criterion.^[2] The key advantage of Baer's criterion lies in its reduction of the injectivity test to ideals alone, bypassing the need to consider all possible submodules of arbitrary modules, which can be computationally intensive or structurally complex. This makes it an indispensable tool for explicit verifications in both theoretical and applied settings.^[5]

Basic Properties and Constructions

Homological Characterization

In homological algebra, an R-module E is injective if and only if the functor \Hom_R(-, E) is exact, meaning that for every short exact sequence $0 \to M \to N \to P \to 0 of R-modules, the induced sequence

$0 \to \Hom_R(P, E) \to \Hom_R(N, E) \to \Hom_R(M, E)

is also exact.^[1] Since the functor \Hom_R(-, E) is always left exact, this condition simplifies to the requirement that \Hom_R(N, E) \to \Hom_R(M, E) is surjective whenever M \to N is injective.^[1] Equivalently, E is injective if and only if \Ext^1_R(K, E) = 0 for every R-module K.^[1] This homological characterization contrasts with that of projective modules, for which the covariant functor \Hom_R(P, -) is exact. In the opposite category, injective modules correspond to projective objects, highlighting the duality between these notions in abelian categories. Injective modules play a central role in homological algebra by enabling the construction of injective resolutions, which dually terminate projective resolutions and facilitate the computation of right derived functors such as \Ext. For instance, over any ring R, the regular module R_R is projective but not necessarily injective, with injectivity holding if and only if R is semisimple.^[6]

Injective Hulls

The injective hull of an R-module M, denoted E(M) or E_R(M), is an injective R-module I together with an injective module homomorphism M \to I such that M is an essential submodule of I.^[7] This means I is the smallest injective module containing M as a submodule, in the sense that any other injective module containing M must contain a submodule isomorphic to I.^[8] An extension M \subseteq I is essential if every nonzero submodule N of I satisfies N \cap M \neq 0.^[9] Thus, the injective hull provides a minimal injective extension where M cannot be enlarged without losing essentiality or injectivity. The existence of the injective hull for every R-module M follows from first embedding M into an injective module I (possible since the category has enough injectives), then applying Zorn's lemma to the partially ordered set of submodules E of I containing M such that the inclusion M \to E is essential, ordered by inclusion; this poset is inductive, and a maximal such E is injective.^[8] The injective hull is unique up to isomorphism: if E(M) and E'(M) are two injective hulls of M, then there exists an isomorphism E(M) \to E'(M) fixing M pointwise.^[10] This uniqueness holds because any two essential extensions into injectives can be composed with homomorphisms extending the identity on M, yielding the isomorphism via maximality. In the category of R-modules, the existence of injective hulls relies on the presence of enough injectives, a property satisfied over any associative ring with identity.^[8] A concrete computation arises for \mathbb{Z}-modules: if M is a torsion-free abelian group (i.e., a torsion-free \mathbb{Z}-module), then its injective hull is E(M) \cong M \otimes_{\mathbb{Z}} \mathbb{Q}.^[11] Here, M embeds densely into the divisible (hence injective) module M \otimes_{\mathbb{Z}} \mathbb{Q}, and the extension is essential since any nonzero element in M \otimes_{\mathbb{Z}} \mathbb{Q} involves a denominator that interacts nontrivially with elements of M.^[11]

Injective Resolutions

An injective resolution of a module M over a ring R is a cochain complex of the form $0 \to M \xrightarrow{d^{-1}} I^0 \xrightarrow{d^0} I^1 \xrightarrow{d^1} \cdots, where each I^i is an injective R-module, the sequence is exact at each I^i for i \geq 0, and the augmented complex is exact (i.e., has vanishing cohomology).^[12] Such resolutions can be constructed iteratively: first embed M into an injective module I^0 (for instance, its injective hull), then embed the cokernel \operatorname{coker}(M \to I^0) into an injective I^1, and continue this process, yielding an exact sequence since the category of R-modules has enough injectives.^[13] Every R-module admits an injective resolution, as the category of left (or right) R-modules is an abelian category with enough injective objects, allowing the iterative embedding construction to terminate in a resolution.^[13] These resolutions are unique up to homotopy equivalence: if I^\bullet and J^\bullet are two injective resolutions of M, then there exists a quasi-isomorphism between them that is homotopic to the identity on M, ensuring that the derived category representation of M is well-defined.^[12] Injective resolutions are fundamental for computing right derived functors, particularly the \operatorname{Ext} groups: for modules M and N, \operatorname{Ext}^n_R(M, N) is isomorphic to the nth cohomology group of the complex \operatorname{Hom}_R(M, I^\bullet), where I^\bullet is an injective resolution of N, obtained by deleting N from the resolution and applying \operatorname{Hom}_R(M, -).^[12] This approach leverages the exactness of the resolution to derive the long exact sequence for \operatorname{Ext} from short exact sequences of modules. Over a principal ideal domain (PID) such as \mathbb{Z}, injective resolutions are particularly simple, with every module having injective dimension at most 1, meaning the resolution terminates after I^1 \to 0.^[14] For example, the module \mathbb{Z}/n\mathbb{Z} over \mathbb{Z} admits an injective resolution of length exactly 1, embedding into the injective module \mathbb{Q}/\mathbb{Z} with cokernel also injective.^[14]

Examples and Applications

Elementary Examples

Over the integers \mathbb{Z}, the rational numbers \mathbb{Q} provide a fundamental example of an injective module, as \mathbb{Q} is a divisible abelian group and injective \mathbb{Z}-modules coincide precisely with the divisible ones.^[15]^[16] In contrast, \mathbb{Z} itself is not injective, since the \mathbb{Z}-homomorphism f: 2\mathbb{Z} \to \mathbb{Z} defined by f(2z) = z cannot be extended to a homomorphism from \mathbb{Z} to \mathbb{Z}, as no element x \in \mathbb{Z} satisfies $2x = 1.^[17] Similarly, for a prime p, the cyclic group \mathbb{Z}/p\mathbb{Z} is not an injective \mathbb{Z}-module, as it is not divisible—for instance, there is no element whose multiple by p yields the generator $1 \mod p.^[3]^[18] To verify the injectivity of \mathbb{Q} explicitly using Baer's criterion, consider any ideal I = n\mathbb{Z} of \mathbb{Z} (with n \geq 0) and any \mathbb{Z}-homomorphism f: n\mathbb{Z} \to \mathbb{Q}. Such an f is determined by f(n) = q for some q \in \mathbb{Q}, and it extends to a homomorphism g: \mathbb{Z} \to \mathbb{Q} by setting g(1) = q/n and extending \mathbb{Z}-linearly, which satisfies g(n \cdot 1) = n \cdot (q/n) = q = f(n).^[5]^[15] This construction works because division by n is always possible in \mathbb{Q}, confirming that \mathbb{Q} satisfies Baer's criterion and is thus injective.^[2] A broader class of examples arises over fields: any vector space over a field k is an injective k-module, as the category of k-modules has global dimension zero, making all modules both projective and injective.^[18]^[2] Finally, over a semisimple Artinian ring R, every R-module is injective, since all modules are semisimple and semisimple modules over such rings are precisely the injective ones.

Commutative and Noetherian Cases

Over a commutative Noetherian ring R, every injective R-module decomposes uniquely (up to isomorphism) as a direct sum of indecomposable injective modules, each of which is the injective hull E(R/\mathfrak{p}) of the residue field at a prime ideal \mathfrak{p} of R.^[19] When R is a principal ideal domain (PID), the structure simplifies further. The indecomposable injectives consist of the quotient field Q of R (the hull of R/(0)) and, for each prime element p \in R, the Prüfer module associated to p, which is the direct limit \varinjlim R/p^n R and serves as the hull of R/(p). Thus, every injective R-module is a direct sum of copies of Q and these Prüfer modules. A concrete example arises over the polynomial ring R = k for a field k. The injective hull of the residue field k \cong R/(x) is E(k) = k(x)/k, the module of rational functions modulo polynomials.

Artinian and Self-Injective Modules

Over Artinian rings, the indecomposable injective modules are the injective hulls of the simple modules, and there are only finitely many simple modules up to isomorphism.^[2] Each such indecomposable injective module has finite length, and hence is Noetherian as a module over the Noetherian ring.^[2] General injective modules are arbitrary direct sums of these indecomposables, which may or may not have finite length depending on the cardinality of the sum. For a module M of finite length over an Artinian ring R, the injective hull E_R(M) also has finite length.^[2] It can be computed explicitly using the socle series of the indecomposable injectives or by determining the multiplicities of the simple composition factors in M and embedding into the appropriate direct sum of indecomposable injectives.^[2] In the local Artinian case, for instance, the length of the hull E_R(k) of the residue field k equals the length of R as an R-module.^[2] A ring R is called self-injective if the regular right module R_R is injective.^[20] Commutative examples include group rings k[G] over a field k for finite abelian groups G, and Artinian principal ideal rings. In the more general (possibly non-commutative) setting, examples include group algebras kG for any finite group G, which are self-injective; in particular, group algebras of finite p-groups over fields of characteristic p are self-injective. Another class of examples consists of full matrix rings M_n(D) over a division ring D, which are self-injective for any n \geq 1.^[21] Over an Artinian ring R, there exists a minimal injective cogenerator, namely the direct sum of the injective hulls of all simple R-modules (one copy for each isomorphism class); since there are finitely many simples, this cogenerator has finite length.^[22]

Modules over Lie Algebras

Although the core theory is developed for commutative rings, injective modules over non-commutative algebras like enveloping algebras of Lie algebras are also studied. In the representation theory of Lie algebras, injective modules are studied primarily as left modules over the universal enveloping algebra U(\mathfrak{g}) of a finite-dimensional Lie algebra \mathfrak{g} over a field k, where k may have arbitrary characteristic.^[23] These modules play a key role in homological algebra and the structure of the category of U(\mathfrak{g})-modules, particularly in settings like the BGG category \mathcal{O}, which consists of finitely generated modules that are locally finite over the nilpotent radical of a Borel subalgebra and semisimple over a Cartan subalgebra. Verma modules, defined as induced modules \Delta(\lambda) = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} k_\lambda from one-dimensional representations of a Borel subalgebra \mathfrak{b} with weight \lambda \in \mathfrak{h}^*, are projective objects in category \mathcal{O} but are not injective in general. In contrast, dual Verma modules \nabla(\lambda), defined via coinduction as \nabla(\lambda) = \mathrm{Hom}_k ( \Delta(\lambda), k ) with the contragredient action, serve as costandard modules and form the building blocks for injectives; every indecomposable injective module in \mathcal{O} admits a finite filtration with subquotients isomorphic to dual Verma modules.^[24] For finite-dimensional semisimple \mathfrak{g} over \mathbb{C}, the injective hulls of simple finite-dimensional modules relate closely to Lie algebra cohomology, where the structure of the injective hull E(L) of a simple module L can be determined via extensions in the cohomology groups H^i(\mathfrak{g}, \mathfrak{n}, L^*), with \mathfrak{n} the nilradical and L^* the dual module.^[23] This connection highlights how injective resolutions contribute to computing Ext-groups and understanding block structures in \mathcal{O}. A concrete example occurs over \mathfrak{sl}(2, \mathbb{C}), where the indecomposable injective modules in category \mathcal{O} are classified by dominant integral highest weights; for each such weight n \geq 0, the injective hull of the simple highest weight module L(n) is the dual Verma module \nabla(-n-2), which has a simple socle isomorphic to L(n) and a Verma filtration dual to that of the projective cover. In modular representation theory, injective modules over the restricted enveloping algebra u(\mathfrak{g}) of a restricted Lie algebra \mathfrak{g} in prime characteristic p exhibit distinct behavior; for instance, every injective u(\mathfrak{g})-module is a direct sum of indecomposables, each with a unique restricted socle, facilitating the study of support varieties and cohomological dimensions in positive characteristic settings.^[25]

Fundamental Theorems

Bass-Papp Theorem

The Bass-Papp theorem addresses the preservation of injectivity under certain module operations, providing both basic closure properties and a characterization of Noetherian rings. For an injective module E over a ring R, a submodule K \subseteq E is injective if and only if K is a direct summand of E. Similarly, a quotient E/K is injective if and only if the short exact sequence $0 \to K \to E \to E/K \to 0 splits, making E/K a direct summand of E. These equivalences follow from the characterization of injective modules via Baer's criterion, which states that a module is injective precisely when it is a direct summand in every super-module containing it as a submodule. Arbitrary products of injective R-modules are always injective. This holds because the functor \Hom_R(-, E) is exact for any injective E, and it converts products of modules into direct sums in the Hom groups, preserving exactness. Finite direct sums of injective modules are also always injective, as they embed as direct summands into the corresponding product. However, arbitrary direct sums of injective modules need not be injective in general. The Bass-Papp theorem states that a ring R is left Noetherian if and only if every direct sum of injective left R-modules is injective. The proof in one direction uses the fact that over Noetherian rings, injective modules admit a decomposition into direct sums of indecomposable injectives, allowing direct sums to remain injective via essential extensions and Baer's criterion. The converse relies on showing that if direct sums of injectives are injective, then ascending chains of annihilator ideals stabilize, implying the Noetherian condition. As a corollary, the category of injective R-modules is closed under arbitrary products for any ring R, but closed under arbitrary direct sums if and only if R is Noetherian. Over Noetherian rings, therefore, injective modules are closed under all direct sums, with finite direct sums sufficing in the trivial sense but arbitrary ones following from the theorem.

Structure Theorem for Injective Modules

Over a commutative Noetherian ring R, the structure theorem for injective modules asserts that every injective R-module I decomposes uniquely as a direct sum I \cong \bigoplus_{\mathfrak{p} \in \operatorname{Spec} R} E(R/\mathfrak{p})^{\kappa_{\mathfrak{p}}}, where the sum is over the prime ideals \mathfrak{p} of R, each E(R/\mathfrak{p}) is the injective hull of the cyclic module R/\mathfrak{p}, and each \kappa_{\mathfrak{p}} is a cardinal (possibly infinite).^[19] This decomposition is essential and unique up to isomorphism and permutation of summands, reflecting the Krull-Schmidt property for injective modules over such rings.^[19] The proof relies on first establishing that every injective module over a commutative Noetherian ring is a direct sum of indecomposable injectives, with the annihilator ideal of each indecomposable summand being prime.^[19] The associated primes of the injective module determine the primes \mathfrak{p} appearing in the decomposition, as each indecomposable E(R/\mathfrak{p}) has annihilator \mathfrak{p}.^[19] Uniqueness follows from the fact that the endomorphism ring of each E(R/\mathfrak{p}) is a local ring (specifically, the completion of the localization R_{\mathfrak{p}} at \mathfrak{p}), ensuring the Krull-Schmidt theorem applies to yield a unique decomposition into these indecomposables.^[2] The indecomposable injective modules in this decomposition are precisely the injective hulls E(R/\mathfrak{p}), which are the minimal injective extensions containing R/\mathfrak{p} as an essential submodule.^[19] These hulls capture the "local" injective structure at each prime, and their direct summands build the global injective. The multiplicities \kappa_{\mathfrak{p}} are determined via Matlis duality, which pairs each injective with a Noetherian module whose structure encodes the cardinalities \kappa_{\mathfrak{p}} through dimensions in the dual category.^[26] In the non-commutative setting, for left Noetherian rings, injective modules similarly decompose uniquely as direct sums of indecomposable injectives, but the indecomposables are injective hulls of simple modules only in restricted cases, such as principal ideal rings.^[19]

Injective Cogenerators

In the category of modules over a ring R, an injective cogenerator is an injective module C such that every R-module embeds into a direct product of copies of C. This means that for any R-module M, there exists a monomorphism M \to \prod_{\lambda \in \Lambda} C_\lambda where each C_\lambda \cong C. Equivalently, C is a cogenerator (i.e., \Hom_R(M, C) \neq 0 for every nonzero M) and is injective.^[27] Module categories always admit injective cogenerators, constructed as the direct sum of the injective hulls of all simple modules; this follows from the existence of enough injectives in abelian categories like R-Mod. The presence of an injective cogenerator implies the existence of injective hulls for all modules, as the essential extension can be realized within products involving the cogenerator. A minimal injective cogenerator is one with no proper injective cogenerator submodule. Over a commutative Noetherian ring R, the minimal injective cogenerator is E = \bigoplus_{m \in \MaxSpec(R)} E_R(R/m), where E_R(R/m) denotes the injective hull of the residue field R/m at each maximal ideal m.^[27]^[28] A ring R is right Artinian if and only if the right regular module R_R is an injective cogenerator. In this case, R is quasi-Frobenius, meaning it is self-injective and the socle of R_R is essential.^[27] Injective cogenerators play a central role in duality theories, such as Matlis duality over commutative Noetherian rings. Here, the minimal injective cogenerator E induces a contravariant functor \Hom_R(-, E) that establishes a duality between the category of Noetherian R-modules and the category of Artinian R-modules, preserving properties like finite length and reflexivity. A module M is reflexive if the natural map M \to \Hom_R(\Hom_R(M, E), E) is an isomorphism, which holds for finitely generated modules when R/\ann(M) is complete semilocal.^[28]^[27]

Advanced Topics

Indecomposable Injective Modules

An indecomposable injective module over a ring R is a nonzero injective R-module that cannot be expressed as a direct sum of two nonzero injective submodules. This property ensures that such modules serve as the basic building blocks in the direct sum decomposition of arbitrary injective modules over suitable rings.^[2] Over a commutative Noetherian ring R, the indecomposable injective modules admit a complete classification: they are precisely the injective hulls E_R(R/\mathfrak{p}) of the residue fields R/\mathfrak{p} at prime ideals \mathfrak{p} of R. This classification arises from the structure theorem for injective modules and highlights the correspondence between the spectrum of R and its indecomposable injectives.^[29]^[2] The endomorphism ring of an indecomposable injective module is always local, meaning its non-units form an ideal. Conversely, an injective module is indecomposable if and only if its endomorphism ring is local. This local property facilitates the unique decomposition of injectives into indecomposables up to isomorphism in appropriate settings.^[30] Over non-Noetherian rings, the classification and structure of indecomposable injective modules become more complex, as exemplified by the indecomposables arising in Baer's foundational work on injectivity criteria for arbitrary rings. For instance, over the integers \mathbb{Z}, the Prüfer p-group \mathbb{Z}(p^\infty) is an indecomposable injective module for each prime p.^[2]

Change of Rings

Consider a ring inclusion \iota: S \hookrightarrow R. The restriction of scalars functor \operatorname{Res}_\iota: {}_R\operatorname{Mod} \to {}_S\operatorname{Mod} views an R-module as an S-module by restricting the action through \iota. If M is an injective R-module, then M viewed as an S-module is S-injective if R is flat as an S-module; in general, without flatness, it need not be.^[31] Now consider a ring homomorphism \phi: R \to S. The extension of scalars functor \operatorname{Ext}_\phi = S \otimes_R -: {}_R\operatorname{Mod} \to {}_S\operatorname{Mod} is left adjoint to the restriction functor \operatorname{Res}_\phi: {}_S\operatorname{Mod} \to {}_R\operatorname{Mod}, satisfying the adjunction

\operatorname{Hom}_S(S \otimes_R N, P) \cong \operatorname{Hom}_R(N, \operatorname{Res}_\phi P)

for R-modules N and S-modules P. This adjunction implies that short exact sequences in {}_R\operatorname{Mod} remain exact after applying the extension functor if S is flat over R, preserving properties like exactness of Hom functors relevant to injectivity.^[31] If M is an injective R-module and S is flat over R, then the extended module S \otimes_R M is an injective S-module. The converse holds if S is faithfully flat over R: M is injective over R if and only if S \otimes_R M is injective over S. This preservation under flat base change follows from the compatibility of injective resolutions with tensor products when torsion vanishes.^[32]^[33] A classical example illustrates the behavior under base change: the rationals \mathbb{Q} form an injective \mathbb{Z}-module, as they are divisible. However, under the extension of scalars via \mathbb{Z} \to \mathbb{Q}, \mathbb{Q} \otimes_\mathbb{Z} \mathbb{Q} \cong \mathbb{Q} becomes a 1-dimensional vector space over the field \mathbb{Q}, and every vector space over a field is both projective and injective.^[2]

Self-Injective Rings

A self-injective ring R is defined as a ring such that R is injective as a right R-module and as a left R-module, i.e., both R_R and _RR are injective modules. This condition implies that the injective dimension of R as a module over itself is zero on both sides.^[20] In a self-injective ring, the regular module R_R (or _RR) is both projective and injective, reflecting the symmetric nature of the ring's module structure over itself. If R is Artinian and self-injective, then R is a quasi-Frobenius ring, characterized by having finite length as a module over itself and admitting a duality between injective and projective modules.^[34] The Nakayama automorphism provides a characterization of self-injectivity in the context of Frobenius structures: for a self-injective algebra \Lambda over a field equipped with a non-degenerate associative bilinear form (-, -): \Lambda \times \Lambda \to k, there exists an automorphism \mu: \Lambda \to \Lambda such that (y, x) = (\mu(x), y) for all x, y \in \Lambda, which induces the duality preserving the module categories. Symmetric algebras form a special case where the Nakayama automorphism is the identity, corresponding to a symmetric bilinear form.^[35]^[36] Examples of self-injective rings include symmetric algebras, such as matrix rings over division rings with the trace form. Another class consists of group rings kG, where G is a finite group and k is a field whose characteristic divides |G|; these are Frobenius algebras and hence self-injective.^[35]^[37]

Generalizations

Injective Objects in Categories

In an abelian category \mathcal{A}, an object I is called injective if for every monomorphism A \to B in \mathcal{A}, the induced map \Hom_{\mathcal{A}}(B, I) \to \Hom_{\mathcal{A}}(A, I) is surjective; equivalently, the contravariant Hom functor \Hom_{\mathcal{A}}(-, I): \mathcal{A}^{\mathrm{op}} \to \mathrm{Ab} is exact.^[38] This generalizes the notion from the category of modules over a ring, where injectivity corresponds to the exactness of \Hom_R(-, I) on short exact sequences. Examples of injective objects abound in certain abelian categories but are absent in others. In the category of sheaves of \mathcal{O}_X-modules on a scheme X, injective sheaves exist and play a key role in computing sheaf cohomology; for instance, the category has enough injectives, allowing every sheaf to be embedded into an injective one.^[39] By contrast, in the category of groups (with group homomorphisms), the only injective object is the trivial group, as any non-trivial group fails the lifting property for certain monomorphisms, such as inclusions of cyclic subgroups. This highlights that not all abelian categories possess non-trivial injectives, unlike the module category over \mathbb{Z}, where divisible groups serve as injectives. A generalization of Baer's criterion provides a practical test for injectivity in broader abelian settings. In a Grothendieck abelian category (which has a small cogenerator), an object I is injective if and only if every morphism from a monomorphism with domain a subobject of the cogenerator extends to the whole object; more precisely, if \{C_i\} is a small set of objects cogenerating \mathcal{A}, then I is injective precisely when \Hom(-, I) is exact on all monomorphisms into the C_i. This criterion, extending the module case where ideals of the ring are tested, enables the construction of injective resolutions in categories like sheaves without relying on the full definition. Abelian categories equipped with enough injectives—meaning every object admits a monomorphism into an injective object—admit injective resolutions, which are essential for defining right derived functors and computing Ext groups. For example, the category of quasi-coherent sheaves on any scheme has enough injectives, facilitating resolutions that compute cohomology; this applies particularly to coherent sheaves on noetherian schemes, where such resolutions yield finite-length complexes in many cases. The existence of enough injectives is guaranteed in Grothendieck categories via the generalized Baer's criterion, ensuring broad applicability.^[40] Historically, the concept of injective objects extends module theory to arbitrary abelian categories through Grothendieck's foundational work in his 1957 Tohoku paper, where he introduced abelian categories and their homological properties. This framework was further generalized to toposes by Lawvere and Tierney in the 1970s, where injective objects in sheaf toposes enable sheaf-theoretic cohomology, and to derived categories by Verdier in his 1960s thesis, allowing injective resolutions to model hyperhomology in triangulated settings.

Divisible Groups

An abelian group G is called divisible if for every element g \in G and every positive integer n \in \mathbb{N}, there exists an element h \in G such that nh = g.^[41] This property ensures that multiplication by n is surjective on G. Over the ring \mathbb{Z} of integers, the injective modules are precisely the divisible abelian groups.^[41] Specifically, an abelian group A is injective as a \mathbb{Z}-module if and only if it is divisible.^[16] This equivalence follows from Baer's criterion applied to the principal ideal domain \mathbb{Z}, where injectivity means exactness of Hom functors for monomorphisms into A. Every divisible abelian group admits a complete structural description: it is isomorphic to a direct sum of copies of the rational numbers \mathbb{Q} (the torsion-free part) and the Prüfer p-groups \mathbb{Z}(p^\infty) for various primes p (the torsion part).^[42] The Prüfer p-group \mathbb{Z}(p^\infty) consists of all p-power roots of unity in the complex numbers, or equivalently, the direct limit of the cyclic groups \mathbb{Z}/p^k\mathbb{Z} as k increases, and it is indecomposable and divisible.^[41] A prominent example is the quotient group \mathbb{Q}/\mathbb{Z}, which is injective as a \mathbb{Z}-module since it decomposes as the direct sum \bigoplus_p \mathbb{Z}(p^\infty) over all primes p.^[42] More generally, any torsion divisible abelian group is a direct sum of Prüfer p-groups \mathbb{Z}(p^\infty), possibly with multiplicities for each p.^[42] For any abelian group A, there exists a divisible hull, which is the smallest divisible subgroup of some injective hull containing A as a dense subgroup, and this hull is itself injective.^[41] This construction embeds A into an injective \mathbb{Z}-module, highlighting the role of divisible groups in completions.

Pure Injective Modules

A module M over a ring R is pure injective if, for every pure monomorphism f: A \hookrightarrow B (meaning the sequence $0 \to A \to B is pure-exact, or equivalently, remains exact after tensoring with any module), every homomorphism g: A \to M extends to a homomorphism h: B \to M such that h \circ f = g. This condition weakens the full injectivity requirement, which demands extensions over all monomorphisms, by restricting to pure monomorphisms—those preserving direct limits of projective resolutions in a specific way. Pure injectivity thus captures a form of "partial injectivity" suited to exactness properties weaker than full exactness, making it particularly useful for studying extensions that are not exact but pure-exact. Every injective module is pure injective, as pure monomorphisms form a subclass of monomorphisms, so the extension property holds automatically. However, the converse fails in general: there exist pure-injective modules that are not injective. Over commutative rings, this distinction is evident; for instance, the ring of p-adic integers \mathbb{Z}_p (for a prime p) is pure injective over \mathbb{Z} but not injective, since it is not divisible. Over non-commutative rings, the gap between the two classes can be similarly pronounced, with additional structural complexities arising from the ring's non-commutativity. This separation highlights pure injectivity as a proper weakening tailored to non-exact extensions.^[44]^[45] Representative examples over \mathbb{Z} illustrate pure injectivity through algebraically compact modules, where pure-injective modules coincide with those satisfying the algebraic compactness condition (solvability of systems of linear equations with finitely many solutions). The p-adic integers \mathbb{Z}_p exemplify this, as do direct products of Prüfer p-groups \prod \mathbb{Z}(p^\infty), which embed pure-injectively while exhibiting compactness properties. Over Noetherian rings, indecomposable pure-injective modules admit a classification via the Ziegler spectrum, associating each to a point in the space of prime theories, though they do not coincide with indecomposable injectives in general. For principal ideal domains (PIDs), pure-injective modules relate closely to divisible (injective) ones but extend to include completions at maximal ideals, such as p-adic modules over \mathbb{Z}.^[44]^[44]^[44] A key structural fact is the existence of pure-injective hulls: for any module N, there is a minimal pure-injective extension, the pure-injective hull \mathrm{PE}(N), which is unique up to isomorphism and contains N as an essential submodule with respect to pure monomorphisms. This mirrors the role of injective hulls but respects the weaker purity condition. In modern applications, pure-injective modules are central to model theory, as in Shelah's classification theorem, which asserts that every consistent theory of modules admits models elementarily equivalent to direct sums of indecomposable pure-injectives, facilitating the study of stable and superstable theories in module categories.^[44]^[44]

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