Definition 4

Let $M$ be an $A$-module. An injective envelope of $M$ is a pair $(I,i)$, where $I$ is an injective $A$-module and $i:M\to I$ is a homomorphism possessing the following property:

(E) for a submodule P of I to be zero, it is necessary and sufficient that i^{-1}(P) be zero.

Note that (E) implies that i is injective. One often identifies M with the submodule i(M) of I, and one then says that I is an injective envelope of M.