Corollary 2

Let A be a ring, and let $\mathfrak{a}$ be a nilpotent two-sided ideal of A. If P is a projective A-module, then $P/\mathfrak{a}P$ is a projective module over $A/\mathfrak{a}$, and the A-module P is indecomposable if and only if the $A/\mathfrak{a}$-module $P/\mathfrak{a}P$ is indecomposable.

Let P be a projective A-module, and let $\overline{P}$ be the $A/\mathfrak{a}$-module $P/\mathfrak{a}P$. The A-module P is zero if and only if $\overline{P}$ is zero (Theorem 2 of VIII, p. 158). Now suppose $P\not= 0$. Since the $A/\mathfrak{a}$-module P is isomorphic to $\overline{A}\otimes_AP$, it is projective (II, §5, No. 1, p. 279, Corollary of Proposition 4). If P is indecomposable, then so is $\overline{P}$ by Corollary 1. Conversely, suppose that P is decomposable and nonzero, and let $P'$ and $P''$ be two nonzero submodules of P such that $P = P'\oplus P''$. By Nakayama’s lemma (VIII, p. 158, Theorem 2), we have $P'+\mathfrak{a}P\not= P$ and $P''+\mathfrak{a}P\not= P$; if $\overline{P}'$ and $\overline{P}''$ are the canonical images of $P'$ and $P''$ in P, then we have $\overline{P}'\not= P,\overline{P}''\not= P$, and $\overline{P}=\overline{P}'\oplus \overline{P}''$. This proves that $\overline{P}$ is decomposable.