Let $R$ be a unital ring, not necessarily commutative, and let $P$ be a right $R$-module. For every integer $n \ge 1$, regard $R^n$ as the free right $R$-module of column vectors with componentwise right scalar multiplication, and let a matrix $p \in M_n(R)$ act on $R^n$ by left multiplication. Then the following conditions are equivalent.

1. The right $R$-module $P$ is finitely generated and projective. 2. There exist an integer $n \ge 1$ and a right $R$-module $Q$ such that $P \oplus Q \cong R^n$ as right $R$-modules. 3. There exist an integer $n \ge 1$ and an idempotent matrix $p \in M_n(R)$, meaning $p^2=p$, such that $P \cong \operatorname{im}(p:R^n\to R^n)$ as right $R$-modules.