Let $A$ be a commutative unital ring and $M$ an $A$-module. Suppose that $M\oplus A \cong A\oplus A$. Then is $M\cong A$?
We have that both $M\oplus A$ and $A\oplus A$ are biproduct for $(A, A)$ and $(M, A)$, so actually short exact sequences. Further $A\oplus A$ is the free $A$-module of rank two. However I can't conclude. Maybe I have to use some extension stuff from homological theory or there are some apparent counter-examples that I'm not able to figure out.