For a ring, let Mod be the category of -modules.
An injective module over is an injective object in .
This is the dual notion of a projective module.
Let be a commutative ring and the category of -modules. We discuss injective modules over (see there for more).
Sketch of proof
Let be a monomorphism in , and let be a map. We must extend to a map . Consider the poset whose elements are pairs where is an intermediate submodule between and and is an extension of , ordered by if contains and extends . By an application of Zorn's lemma, this poset has a maximal element, say . Suppose is not all of , and let be an element not in ; we show that extends to a map , a contradiction.
The set is an ideal of , and we have a module homomorphism defined by . By hypothesis, we may extend to a module map . Writing a general element of as where , it may be shown that
is well-defined and extends , as desired.
By Baer’s criterion, it suffices to show that for any ideal of , a module homomorphism extends to a map . Since is Noetherian, is finitely generated as an -module, say by elements . Let be the projection, and put . Then for each , is nonzero for only finitely many summands. Taking all of these summands together over all , we see that factors through
for some finite . But a product of injectives is injective, hence extends to a map , which completes the proof.
This is due to Bass and Papp. See (Lam, Theorem 3.46).
Existence of enough injectives
We discuss that in the presence of the axiom of choice at least, the category Mod has enough injectives in that every module is a submodule of an injective one. We first consider this for . We do assume prop. 4, which may be proven using Baer's criterion.
By prop. 4 an abelian group is an injective -module precisely if it is a divisible group. So we need to show that every abelian group is a subgroup of a divisible group.
To start with, notice that the group of rational numbers is divisible and hence the canonical embedding shows that the additive group of integers embeds into an injective -module.
Now by the discussion at projective module every abelian group receives an epimorphism from a free abelian group, hence is the quotient group of a direct sum of copies of . Accordingly it embeds into a quotient of a direct sum of copies of .
Here is divisible because the direct sum of divisible groups is again divisible, and also the quotient group of a divisible groups is again divisble. So this exhibits an embedding of any into a divisible abelian group, hence into an injective -module.
The proof uses the following lemma.
Write for the forgetful functor that forgets the -module structure on a module and just remembers the underlying abelian group .
The functor has a right adjoint
given by sending an abelian group to the abelian group
equipped with the -module struture by which for an element is sent to the element given by
This is called the coextension of scalars along the ring homomorphism .
The unit of the adjunction
is the -module homomorphism
given on by
of prop. 3
Let . We need to find a monomorphism such that is an injective -module.
By prop. 2 there exists a monomorphism
of the underlying abelian group into an injective abelian group .
Now consider the adjunct of , hence the composite
with and from lemma 1. On the underlying abelian groups this is
Once checks on components that this is a monomorphism. Therefore it is now sufficient to see that is an injective -module.
This follows from the existence of the adjunction isomorphism given by lemma 1
natural in and from the injectivity of .
Injective -modules / abelian groups
Let Ab be the abelian category of abelian groups.
Using Baer’s criterion, prop. 1.
Not injective in Ab is for instance the cyclic group for .
The notion of injective modules was introduced in
(The dual notion of projective modules was considered explicitly only much later.)
A general discussion can be found in
The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.
Baer’s criterion is discussed in many texts, for example
- N. Jacobsen, Basic Algebra II, W.H. Freeman and Company, 1980.
- T.-Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics 189, Springer Verlag (1999).
Section 4.2 of
For abelian sheaves over the etale site: