Exercise 4, § 5

Let A and B be rings and P an $(A$, B)-bimodule.

a) Suppose that the B-module P is generating and that the A-module P is balanced. Show that for every balanced B-module M, the A-module $P\otimes_BM$ is balanced.

b) Suppose that the A-module P is generating and that the mapping $b\mapsto b_P$ from B to End$_A(P)$ is bijective. Prove that for every generating B-module M, the A-module $P\otimes_BM$ is generating.