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.
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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.
