Injective Module
An injective module is the dual notion to the projective module. A module 
over a unit
ring 
is called injective iff
whenever 
is contained as a submodule
in a module 
, there exists
a submodule 
of 
such that the
direct sum 
is
isomorphic to 
(in other words, 
is a direct
summand of 
). The subset 
of 
is an example
of a noninjective 
-module; it is a 
-submodule of 
, and it is isomorphic to 
; 
, however, is
not isomorphic to the direct sum 
.
The field of rationals 
and its quotient
module 
are examples of injective 
-modules.
A direct product of injective modules is always injective. The corresponding property for direct sums does not hold in general, but it is true for modules over Noetherian
rings.
The notion of injective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors.
SEE ALSO: Commutative Diagram,
Baer's Criterion,
Cofree
Module,
Divisible Module,
Exact
Functor,
Projective Module
This entry contributed by Margherita
Barile
REFERENCES:
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Injective Module
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