Corollary 4

Every finitely generated A-module is of finite rank.

Note that the converse of this corollary is not necessarily true; for example $\mathbf{Q}$ is a $\mathbf{Z}$-module of rank 1 but is not finitely generated over $\mathbf{Z}$.

Recall (§ 5, no. 1) that for every linear mapping $f : E \to E'$ (where $E$ and $E'$ are $A$-modules), $f_{(K)}$ denotes the $K$-linear mapping $1_K \otimes f : E_{(K)} \to E'_{(K)}$.