Let $R$ be a commutative Ring with identity. We have an Artinian $R$-Module $M$ and an $R$-Homomorphism $$ \phi: M \rightarrow M$$ which is injective. My task is it to show, that $\phi$ is surjective, hence $\phi$ is an Automorphism.
(My idea was: To use the fact that $M$ is Artinian I would like to construct a descanding chain of sub-structures of $M$, furthermore I would like to bring this into relation with $\phi$ should being surjective. Like playing around with the cokernel of $\phi$.)
I would be thankful for any kind of help.