Corollary 3

Every finitely generated module over a local ring has a projective cover.

Let A be a local ring and $\mathfrak{r}$ its radical. It is a two-sided ideal of A (VIII, p. 155, Proposition 5, a)), and the ring $A/\mathfrak{r}$ is a field (VIII, p. 157, Corollary 4). If M is an A-module, then $M/\mathfrak{r}M$ is a vector space over the field $A/\mathfrak{r}$, hence is a free $(A/\mathfrak{r}$)-module. It then suffices to apply Corollary 2.