Corollary

Let P and $P'$ be projective A-modules. If the modules $P/\mathfrak{a}P$ and $P'/\mathfrak{a}P'$ are isomorphic, then P and $P'$ are isomorphic.

Since $P/\mathfrak{a}P$ and $P'/\mathfrak{a}P'$ are projective, the corollary follows from Proposition 11 and the uniqueness of the projective cover (VIII, p. 162, Corollary 1).