Injection and surjection over free modules. 
Let $A$ be a commutative ring and $M$ an $A$-module.
  Suppose to have both an injection $A^s \to M$ and a surjection $A^s \to M$ of module homomorphisms. Show that $M \simeq A^s$.

This point is the last thing that I need to conclude another homework I started two days ago (An exercise on tensor product in a local integral domain.).
I am curious to know if it is true in that context. Furthermore, also a proof for the easier case $A$ local integral domain (as indicated in my previous exercise) should be absolutely good (because it will lead me to solve the exercise).
Thank you in advance, any suggestion will be appreciated.

Let $\sigma:R^s\to M$ be an injective homomorphism, and $\pi:R^s\to M$ a surjective homomorphism. Set $N=\sigma(R^s)$ and notice that $\sigma:R^s\to N$ is an isomorphism. Now consider $\pi\sigma^{-1}:N\to M$. This is a surjective homomorphism, hence by Orzech Theorem an isomorphism.
