Characterization of free modules

Question

Let $M$ be a finitely generated module over a commutative ring $A$. Is it true that if there exists a positive integer $n$ and a pair of homomorphisms $\pi:A^n\rightarrow M$ and $\phi:A^n\rightarrow M$ such that $\pi$ is surjective and $\phi$ is injective then $M\cong A^n$?

Answer 1

Yes, it is.

Let $N=\phi(A^n)$. Then $N$ is a submodule of $M$ isomorphic to $A^n$. Now define $f:N\to M$ by $f(\phi(x))=\pi(x)$. Then $f$ is surjective, and therefore injective. We thus get that $M$ is free of rank $n$.