Injective object

Definition

More formally, a left module Q over the ring R is injective if it satisfies one (and therefore all) of the following equivalent conditions:

• 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 internal direct sum of Q and K, i.e. Q + K = M and Q ∩ K = {0}.
• If X is a submodule of the left R-module Y and g : X → Q is a module homomorphism, then there exists a module homomorphism h : Y → Q such that h(x) = g(x) for all x in X.
• If X and Y are left-R modules and f : X → Y is an injective 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 commutes:

• Any short exact sequence 0 →Q → M → K → 0 of left R-modules splits.
• The contravariant functor Hom(-,Q) from the category of left R-modules to the category of abelian groups is exact.

Injective right R-modules are defined in complete analogy.

Examples

Trivially, the zero module {0} is injective.

Every vector space Q is injective. Reason: if V is a subspace of W, we can find a basis of V and extend it to a basis of W. A given linear map from V to Q can then be extended to W by keeping it fixed on the basis of V, defining it arbitrarily on the rest of the basis, and extending it to all of W by linearity. This example shows in particular that the extending map (called g in the above definition) is typically not unique.

If G is a finite group and k a field with characteristic 0, then the group algebra kG has the property that all short exact sequences of kG modules split. As a consequence, all kG modules are injective in this case. 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 Hom_k(−, 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 Hom_k(P, k) where P is a finitely generated projective right A-module.

Facts

Any product of injective modules is injective. In general, submodules, factor modules or infinite direct sums of injective modules need not be injective.

To show that a given module is injective, the following Injective Test Lemma is useful: 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. One may view the example about vector spaces as a special case of this theorem, as every vector space is divisible.

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 subgroup of an injective one. It is quite significant that this is true over any ring: every module is a submodule of an injective one. This is also expressed as: "the category of left R-modules has enough injective." 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.

One can then go on to define the injective hull of a module (essentially the smallest injective module containing the given one). Every module M also has an injective resolution: an exact sequences of the form

0 → M → I⁰ → I¹ → I² → ...

Generalization

One also talks about injective objects in categories more general then module categories, for instance in functor categories or in categories of sheaves of O_X modules over some ringed space (X,O_X). The following general definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g.

References

• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992.

Referenced By