Remark 4

Suppose that every finitely generated A-module has a projective cover; then the quotient $A'$ of A by its radical is semisimple. Indeed, every finitely generated module over the ring $A'$ is projective by the corollary. In particular, for every left ideal $\mathfrak{a}$ of $A'$, the $A'$-module $A'_s/\mathfrak{a}$ is projective. Our assertion then follows from Proposition 4 of VIII, p. 138.

We can give examples of a commutative ring A for which $A/\mathfrak{r}$ is semisimple and of a finitely generated A-module M that does not admit a projective cover (VIII, p. 170, Exercise 22).