Remark 3

Suppose that the ring A is without radical. By Proposition 9, $(P, u)$ is a projective cover of an A-module M if and only if $u$ is an isomorphism. Hence, projective covers can only exist for projective A-modules.

The ring $\mathbf{Z}$ is without radical (VIII, p. 154, Example 3). Let $n\geqslant 2$ be an integer. The $\mathbf{Z}$-module $\mathbf{Z}/n\mathbf{Z}$ is not projective and therefore does not admit a projective cover.