Remark 2

Suppose that $(P, u)$ is a projective cover of the A-module M. Let $(x_i)_{i\in I}$ be a family of elements of P, and let $P'$ be the submodule of P that it generates; then $u(P')$ is generated by the family $(u(x_i))_{i\in I}$. Consequently, the family $(x_i)_{i\in I}$ generates the A-module P if and only if the family $(u(x_i))_{i\in I}$ generates the A-module M. In particular, P is finitely generated if and only if M is finitely generated.