Proposition 10

Let M be an A-module, P a projective A-module, and $u: P\rightarrow M$ a linear mapping. Let $\mathfrak{a}$ be a two-sided ideal of A. Suppose that the linear mapping $\overline{u}: P/\mathfrak{a}P\rightarrow M/\mathfrak{a}M$ deduced from $u$ by passing to the quotients is bijective and that one of the following conditions are satisfied:

(i) The A-modules M and P are finitely generated, and $\mathfrak{a}$ is contained in the radical of A.

(ii) The ideal $\mathfrak{a}$ is nilpotent.

Then $(P, u)$ is a projective cover of M.

Under the assumptions, the homomorphism $u$ is surjective (VIII, p. 159, Corollary 2) and its kernel N is contained in $\mathfrak{a}P$. Let $P'$ be a proper submodule of P. By Nakayama’s lemma (VIII, p. 158, Theorem 2), we have $P'+\mathfrak{a}P\not= P$ and therefore $u(P')\not= M$. So $(P, u)$ is a projective cover of M.