Injective modules were introduced in (Baer 1940) and are discussed in some detail in the textbook (Lam 1999, §3).
Reason: if Q is a subspace of V, we can find a basis of Q and extend it to a basis of V. The new extending basis vectors span a subspace K of V and V is the internal direct sum of Q and K. Note that the direct complement K of Q is not uniquely determined by Q, and likewise the extending map h in the above definition is typically not unique.
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
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, (Lam 1999, §3I).
The endomorphism ring of the injective hull of R/P is the completion
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.
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 (Lam 1999, §3G, §3J).
This implies every local Gorenstein ring which is also Artin is injective over itself since has a 1-dimensional socle.
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, (Lam 1999, p. 80-81, Th 3.46).
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).
Baer's criterion has been refined in many ways (Golan & Head 1991, p. 119), including a result of (Smith 1981) and (Vámos 1983) 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.
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.
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 any ring R, a left R-module is flat if and only if its character module is injective.
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.
Every indecomposable injective module has a local endomorphism ring.
Over a commutative Noetherian ring, this gives a particularly nice understanding of all injective modules, described in (Matlis 1958).
The indecomposable injective modules are the injective hulls of the modules R/p for p a prime ideal of the ring R. Moreover, the injective hull M of R/p has an increasing filtration by modules Mn given by the annihilators of the ideals pn, and Mn+1/Mn is isomorphic as finite-dimensional vector space over the quotient field k(p) of R/p to HomR/p(pn/pn+1, k(p)).
The same statement holds of course after interchanging left- and right- attributes.
In particular, if R is an integral domain and S its field of fractions, then every vector space over S is an injective R-module.
makes R into a left-R, right-S bimodule, by left and right multiplication.
Specializing the above statement for P = R, it says that when M is an injective right S-module the coinduced module
Applying this to R=Z, I=nZ and M=Q/Z, one gets the familiar fact that Z/nZ is injective as a module over itself.
The textbook (Rotman 1979, p. 103) has an erroneous proof that localization preserves injectives, but a counterexample was given in (Dade 1981).
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. The notion of injective object in the category of abelian groups was studied somewhat independently of injective modules under the term divisible group.
In relative homological algebra, the extension property of homomorphisms may be required only for certain submodules, rather than for all.