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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).
If the axiom of choice holds, then a module is an injective module precisely if for any left -ideal regarded as an -module, any homomorphism in can be extended to all of along the inclusion .
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.
Assume that the axiom of choice holds.
Let be a Noetherian ring, and let be a collection of injective modules over . Then the direct sum is also injective.
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.
Conversely, is a Noetherian ring if direct sums of injective -modules are injective.
This is due to Bass and Papp. See (Lam, Theorem 3.46).
Given a pair of additive adjoint functors
between abelian categories such that the left adjoint is an exact functor, then the right adjoint preserves injective objects.
Observe that an object in an abelian category is injective precisely if the hom-functor into it sends monomorphisms to epimorphisms (prop.), and that preserves monomorphisms by assumption of exactness. With this the statement follows via adjunction isomorphism
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. , which may be proven using Baer's criterion.
Assuming the axiom of choice, the category Mod Ab has enough injectives.
By prop. 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.
Given a pair of additive adjoint functors
between abelian categories such that the left adjoint is
an exact functor,
Then if has enough injectives, also has enough injectives.
Consider . By the assumption that has enough injectives, there is an injective object and a monomorphism . The adjunct of this is a morphism
and so it is sufficient to show that
is injective in ;
is a monomorphism.
The first point is the statement of lemma .
For the second point, consider the kernel of as part of the exact sequence
By the assumption that is an exact functor, the image of this sequence under is still exact
Now observe that is a monomorphism: this is because its composite with the adjunction unit is (by the formula for adjuncts) the original morphism , which by construction is a monomorphism. Therefore the exactness of the above sequence means that is the zero morphism; and by the assumption that is a faithful functor this means that already is zero, hence that , hence that is a monomorphism.
As soon as the category Ab of abelian groups has enough injectives, so does the abelian category Mod of modules over some ring .
In particular if the axiom of choice holds, then has enough injectives.
Observe that the forgetful functor has both a left adjoint (extension of scalars from to ) and a right adjoint (coextension of scalars). Since it has a left adjoint, it is exact. Thus the statement follows via lemma from prop. .
For a field, hence Mod = Vect, every object is both injective as well as projective.
Let Ab be the abelian category of abelian groups.
An abelian group is injective as a -module precisely if it is a divisible group, in that for all integers we have .
Using Baer’s criterion, prop. .
By prop. the following abelian groups are injective in Ab.
The group of rational numbers is injective in Ab, as is the additive group of real numbers and generally that underlying any field. The additive group underlying any vector space is injective. The quotient of any injective group by any other group is injective.
Not injective in Ab is for instance the cyclic group for .
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution
The notion of injective module 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
See also
Section 4.2 of
For abelian sheaves over the etale site:
A study of injective modules in higher algebra:
Last revised on September 22, 2017 at 08:44:14. See the history of this page for a list of all contributions to it.