Proposition 12

Let E be an injective A-module. In order that E be a cogenerator, it is necessary and sufficient that $\mathrm{Hom}_A(S, E) \neq 0$ for every simple A-module S.

The condition is obviously necessary. Conversely, let M be an A-module and x a nonzero element of M; the submodule $Ax$ of M has a simple quotient S (VIII, § 2, no 1, prop. 3). If $\mathrm{Hom}_A(S, E) \neq 0$, then $\mathrm{Hom}_A(Ax, E) \neq 0$ and there exists a homomorphism $f : Ax \to E$ such that $f(x) \neq 0$; since E is injective, $f$ extends to a homomorphism $u$ of M into E and we have $u(x) = f(x) \neq 0$.