Exercise 20, § 2

Let A be ring, let M be an A-module of finite length, and let N be an A-module. Let $m$ and $n$ be strictly positive integers such that $M^m$ and $N^n$ are isomorphic. Prove that there exist an A-module P and integers $p, q$ such that M is isomorphic to $P^p, N$ is isomorphic to $P^q$, and $mp=nq$.