I try to find a counterexample of the exercise 2.5 in Commutative Ring Theory by Matsumura, when $A$ is not commutative. The exercise says.
Let $A$ be a commutative ring. If $A^n \simeq A^m$, then $m = n$.
I have thought that the module $A^m$ has a minimal base with $n$ elements $\{x_1, ..., x_n\}$ and need to take the $\phi : A^n \rightarrow A^m$ such that $\phi (e_j) = x_j$ where $e_j$ is the canonical base.
