Exercise 23, § 5

Let $A$ be a ring and $(f_i)_{i \in I}$ a finite family of elements of $A$ generating the ideal $A$.

(a) Let $M$ be an $A$-module. Show that, if, for all $i \in I$, $M_{f_i}$ is a finitely generated $A_{f_i}$-module, $M$ is a finitely generated $A$-module.

(b) Let $B$ be a (commutative) $A$-algebra. Show that, if, for all $i \in I$, $B_{f_i}$ is a finitely generated $A_{f_i}$-algebra, $B$ is a finitely generated $A$-algebra.