Proposition 11

Let A be a ring and $\mathfrak{a}$ a nilpotent two-sided ideal of A. Let M be a projective $A/\mathfrak{a}$-module. There exist a projective A-module P and a surjective A-linear mapping $u: P\rightarrow M$ with kernel $\mathfrak{a}P$.

Such a pair $(P, u)$ is a projective cover of M viewed as an A-module.

There exists an $A/\mathfrak{a}$-module $M'$ such that $M\oplus M'$ is a free $A/\mathfrak{a}$-module. Choose a free A-module L and a surjective A-linear mapping $v: L\rightarrow M\oplus M'$ with kernel $\mathfrak{a}L$. By Corollary 1 of Proposition 7 (VIII, p. 159), there exists a direct sum decomposition $L = P\oplus P'$ such that $v(P) = M$ and $v(P') = M'$. The A-module P is projective, and the A-linear mapping $u$ from P to M that coincides with $v$ on P is surjective, with kernel $\mathfrak{a}P$. The first assertion follows.

Let P be a projective A-module and $u$ a homomorphism from P to M with kernel $\mathfrak{a}P$. By Proposition 10, the pair $(P, u)$ is a projective cover of M.