Definition 3

One says that the A-module E is a cogenerator if, for every A-module M and every nonzero element x of M, there exists an A-homomorphism $u : M \to E$ such that $u(x) \neq 0$.

One says that the A-module L is a generator if, for every A-module M and every element x of M, there exists an A-homomorphism $u : L \to M$ such that $x \in u(L)$. For example, the A-module $A_s$ is a generator.