An injective module is the dual notion to the projective module. A module M over a unit ring R is called injective iff whenever M is contained as a submodule in a module N, there exists a submodule X of N such that the direct sum M⊕X is isomorphic to N (in other words, M is a direct summand of N). The subset {0, 2} of Z_4 is an example of a noninjective Z-module; it is a Z-submodule of Z_4, and it is isomorphic to Z_2; Z_4, however, is not isomorphic to the direct sum Z_2 ⊕Z_2. The field of rationals Q and its quotient module Q/Z are examples of injective Z-modules.

Baer's criterion | cofree module | commutative diagram | divisible module | exact functor | projective module