In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook .
Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators 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 injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable 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 group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.
Definition
A left module over the ring is injective if it satisfies one (and therefore all) of the following equivalent conditions: If is a submodule of some other left -module , then there exists another submodule of such that is the internal direct sum of and , i.e. and . Any short exact sequence of left -modules splits. If and are left -modules, is an injective module homomorphism and is an arbitrary module homomorphism, then there exists a module homomorphism such that , i.e. such that the following diagram commutes:
The contravariant Hom functor from the category of left -modules to the category of abelian groups is exact.
Injective right -modules are defined analogously.
Examples First examples
Trivially, the zero module is injective.
Given a field , every -vector space is an injective -module. Reason: if is a subspace of , we can find a basis of and extend it to a basis of . The new extending basis vectors span a subspace of and is the internal direct sum of and . Note that the direct complement of is not uniquely determined by , and likewise the extending map in the above definition is typically not unique.
The rationals (with addition) form an injective abelian group (i.e. an injective -module). The factor group and the circle group are also injective -modules. The factor group for is injective as a -module, but not injective as an abelian group.
Commutative examples
More generally, for any integral domain 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 Dedekind domain, the quotient module K/R is also injective, and its indecomposable summands are the localizations for the nonzero prime ideals . The zero ideal 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 commutative noetherian rings due to Eben Matlis, . 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 prime spectrum 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 local rings. The endomorphism ring of the injective hull of R/P is the completion of R at P.
Two examples are the injective hull of the Z-module Z/pZ (the Prüfer group), 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 formal power series.
Artinian examples
If G is a finite group and k a field with characteristic 0, then one shows in the theory of group representations 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 group algebra kG are injective. If the characteristic of k is not zero, the following example may help.
If A is a unital associative algebra over the field k with finite dimension over k, then Homk(−, k) is a duality 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 symmetric algebras, the duality is particularly well-behaved and projective modules and injective modules coincide.
For any Artinian ring, just as for commutative rings, 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 injective hull. For finite-dimensional algebras over fields, these injective hulls are finitely-generated modules .
Computing injective hulls If is a Noetherian ring and is a prime ideal, set as the injective hull. The injective hull of over the Artinian ring can be computed as the module . It is a module of the same length as . In particular, for the standard graded ring and , is an injective module, giving the tools for computing the indecomposable injective modules for artinian rings over .
Self-injectivity An Artin local ring is injective over itself if and only if is a 1-dimensional vector space over . This implies every local Gorenstein ring which is also Artin is injective over itself since has a 1-dimensional socle. A simple non-example is the ring which has maximal ideal and residue field . Its socle is , which is 2-dimensional. The residue field has the injective hull .
Modules over Lie algebras For a Lie algebra over a field of characteristic 0, the category of modules has a relatively straightforward description of its injective modules. Using the universal enveloping algebra any injective -module can be constructed from the -modulefor some -vector space . Note this vector space has a -module structure from the injectionIn fact, every -module has an injection into some and every injective -module is a direct summand of some .
Theory
Structure theorem for commutative Noetherian rings Over a commutative Noetherian ring , 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 . That is, for an injective , there is an isomorphismwhere are the injective hulls of the modules . In addition, if is the injective hull of some module then the are the associated primes of .
Submodules, quotients, products, and sums, Bass-Papp Theorem
Any product of (even infinitely many) injective modules is injective; conversely, if a direct product of modules is injective, then each module is injective . Every direct sum of finitely many injective modules is injective. In general, submodules, factor modules, or infinite direct sums of injective modules need not be injective. Every submodule of every injective module is injective if and only if the ring is Artinian semisimple ; every factor module of every injective module is injective if and only if the ring is hereditary, .
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) Noetherian, .This is the Bass-Papp theorem, see and
Baer's criterion
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 left ideal I of R can be extended to all of R.
Using this criterion, one can show that Q is an injective abelian group (i.e. an injective module over Z). More generally, an abelian group is injective if and only if it is divisible. More generally still: a module over a principal ideal domain 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 , including a result of and that for a commutative Noetherian ring, it suffices to consider only prime ideals 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.
Injective cogenerators
Maybe the most important injective module is the abelian group Q/Z. It is an injective cogenerator in the category of abelian groups, 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 projective modules, but between injective modules and flat modules . 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.
Injective hulls
The injective hull of a module is the smallest injective module containing the given one and was described in .
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.
Injective resolutions Every module M also has an injective resolution: an exact sequence of the form 0 → M → I0 → I1 → I2 → ... where the I j are injective modules. Injective resolutions can be used to define derived functors such as the Ext functor.
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. 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 module isomorphic to an injective module is of course injective.