Proposition 5

Let A be a semi-local ring and P a finitely generated projective A-module. If the rank of P is defined, P is a free A-module.

Suppose first that A is isomorphic to a product of fields $K_i$ ($1 \leq i \leq n$). The $K_i$ are then identified with the minimal ideals (Algebra, Chapter VIII, § 3, no. 1) of A and, for all $i$, the sum $p_i$ of the K, of index $j \neq i$ is a maximal ideal of A, the $p_i$ ($1 \leq i \leq n$) being the only prime ideals of A. Every finitely generated A-module P is therefore the direct sum of its isotypical components $P_i$ ($1 \leq i \leq n$), $P_i$ being isomorphic to a direct sum of a finite number $r_i$ of A-modules isomorphic to K, (Algebra, Chapter VIII, § 5, no. 1, Proposition 1 and no. 3, Proposition 11); the ring $A_{r_i}$ is identified with K, and annihilates the $P_j$ of index $j \neq i$, hence $r_i = \mathrm{rg}_{p_i}(P)$; if all the $r_i$ are equal to the same number $r$, P is isomorphic to $A^r$, whence the proposition in this case. In the general case, let $\mathfrak{R}$ be the Jacobson radical of A and $B = A / \mathfrak{R}$; as B is a product of fields, the projective B-module $P_{(B)}$ is free by the remarks preceding Proposition 4. Also P is a flat A-module and the proposition then follows from § 3, no. 2, Proposition 5.