How can I show that any one to one endomorphism of an Artinian module $M$ is an automorphism?
I was given this question and I presume that it is really to show that Artinian modules are co-hopfian. I think I am missing something easy, but I am just stuck where I cannot see why what I have shown so far helps me:
If $f ∈End(M)$ is injective, then we have a descending chain of submodules $Im(f^k)⊇Im(f^{k+1})⊇Im(f^{k+2})⊇.. $ This sequence must be finite due to $M$ being an artinian module.
Is this the right way to begin this question, and if so where do I go from here?
Suppose $n$ is the least +ve integer such that $Im(f^{n+1})=Im(f^n).$ Let $m \in M.$ Choose $m' \in M$ such that $f^n(m)=f^{n+1}(m') = f^n(f(m')).$ Now use the injectivity of $f$ to conclude that $m=f(m').$