An
injective object
in the category of (right) modules over an associative ring with identity
,
i.e. an
-module
such that for any
-modules
,
,
for any monomorphism
,
and for any homomorphism
there is a homomorphism
that makes the following diagram commutative
Here and below all
-modules
are supposed to be right
-modules.
The following conditions on an
-module
are equivalent to injectivity: 1) for any
exact sequence
the induced sequence
is exact; 2) any exact sequence of
-modules
of the form
splits,
i.e. the submodule
is a direct summand of
;
3)
for all
-modules
;
and 4) for any right ideal
of
a homomorphism of
-modules
can be extended to a homomorphism of
-modules
(Baer's criterion).
There are
"enough"
injective objects in the category of
-modules:
Each
-module
can be imbedded in an injective module. Moreover, each module
has an
injective hull
,
i.e. an injective module containing
in such a way that each non-zero submodule of
has non-empty intersection with
.
Any imbedding of a module
into an injective module
can be extended to an imbedding of
into
.
Every
-module
has an
injective resolution
i.e. an exact sequence of modules in which each module
,
,
is injective. The length of the shortest injective resolution is called the
injective dimension
of the module (cf. also
Homological dimension).
A direct product of injective modules is an injective module. An injective module
is equal to
for any
that is not a left zero divisor in
,
i.e. an injective module is divisible. In particular, an
Abelian group is an injective module over the ring
if and only if it is divisible. Let
be a commutative Noetherian ring. Then any injective module
over it is a direct sum of injective hulls of modules of the form
,
where
is a prime ideal in
.
Injective modules are extensively used in the
description of various classes of rings (cf.
Homological classification of rings).
Thus, all modules over a ring are injective if and only
if the ring is semi-simple. The following conditions are equivalent:
is a right Noetherian ring; any direct sum of injective
-modules
is injective; any injective
-module
is decomposable as a direct sum of indecomposable
-modules.
A ring
is right Artinian if and only if every injective module is
a direct sum of injective hulls of simple modules. A ring
is right hereditary if and only if all its quotient modules by injective
-modules
are injective, and also if and only if the
sum of two injective submodules of an arbitrary
-module
is injective. If the ring
is right hereditary and right Noetherian, then every
-module
contains a largest injective submodule. The
projectivity (injectivity) of all injective (projective)
-modules
is equivalent to
being a
quasi-Frobenius ring.
The injective hull of the module
plays an important role in the theory of rings of
fractions. E.g., if the right singular ideal of a ring
vanishes, if
is the injective hull of the module
,
and if
is its endomorphism ring, then the
-modules
and
are isomorphic,
is a ring isomorphic to
and is also the maximal right ring of fractions of
,
and
is a self-injective right
regular ring (in the sense of von Neumann).
In connection with various problems on extending
module homomorphisms, some classes of modules
close to injective modules have been considered:
quasi-injective modules
(if
and
,
then
can be extended to an endomorphism of
);
pseudo-injective modules
(if
and
is a monomorphism, then
can be extended to an endomorphism of
);
and
small-injective modules
(all endomorphisms of submodules can be extended to endomorphisms of
).
The quasi-injectivity of a module
is equivalent to the invariance of
in its injective hull under endomorphisms of the latter.