Injective module
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
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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
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the induced sequence
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is exact; 2) any exact sequence of -modules of the form
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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
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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.
References
[1] | H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) |
[2] | S. MacLane, "Homology" , Springer (1963) |
[3] | C. Faith, "Lectures on injective modules and quotient rings" , Springer (1967) |
[4] | D.W. Sharpe, P. Vamos, "Injective modules" , Cambridge Univ. Press (1972) |
Comments
A ring is called right hereditary if every right ideal is projective or, equivalently, if its right global dimension is . It is called semi right hereditary if every finitely-generated right ideal is projective. Commutative hereditary integral domains are Dedekind rings; a commutative semi-hereditary integral domain is called a Prüfer ring. A right hereditary ring need not be also left hereditary.
References
[a1] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) |
[a2] | J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) pp. Part I, Chapt. 2 |
Injective module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Injective_module&oldid=39556