Definition 2

*Let $n$ be an integer $\geqslant 0$. A projective $A$-module $P$ is said to be of rank $n$ if it is finitely generated and $\mathrm{rg}_p(P) = n$ for every prime ideal $p$ of $A$.*

Clearly every finitely generated free $A$-module $L$ is of rank $n$ in the sense of Definition 2, $n$ being equal to the dimension (or rank) of $L$ defined in Algebra, Chapter II, § 7, no. 2.

A projective module of rank 0 is zero (§ 3, no. 3, Corollary 2 to Theorem 1). If $A$ is not reduced to 0 and a projective $A$-module $P$ is of rank $n$, the integer $n$ is determined uniquely; it is then denoted by $\mathrm{rg}(P)$.