Proposition 9

Let B be a ring, A a subring of B and P a projective left A-module. Then, if $P_{(B)}$ is a finitely generated B-module, P is itself a finitely generated A-module.

We know (§ 2, no. 6, Proposition 12) that there exists a family $(a_\lambda)_{\lambda \in L}$ of elements of P and a family $(a_\lambda^*)_{\lambda \in L}$ of elements of the dual P* such that, for all $x \in P$, the family $\langle x, a_\lambda^* \rangle$ is of finite support and $x = \sum_\lambda \langle x, a_\lambda^* \rangle a_\lambda$. Since $P_{(B)}$ is finitely generated, there exists a finite family $(y_i)_{i \in I}$ of elements of P such that $P_{(B)}$ is generated by the elements $1 \otimes y_i$. For each index i, the family $\langle y_i, a_\lambda^* \rangle$ has finite support. Hence there exists a finite subset H of L such that $\langle y_i, a_\lambda^* \rangle = 0$ for $i \in I$ and $\lambda \notin H$. Since

it follows that $1 d_B \otimes a_\lambda^* = 0$ for $\lambda \notin H$. Hence, for all $x \in P$, for $\lambda \notin H$. This shows that the A-module P is generated by the $a_\lambda$ such that $\lambda \in H$.