Exercise 23, § 5

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

(a) Let M be an A-module. Now prove 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. Now prove 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.