The following question is related to this post: Example of a an endomorphism which is not a right divisor of zero and not onto
I was told there is a simple motivation for this problem from a linear algebra textbook such as Hofmann and Kunze but I have yet to find a clear explanation. In particular I do not see why existence of one invertible element tells us about the behavior of injective endomorphisms... I think this problem is related to exercises in Algebra Volume II by Bourbaki but the notation is probably a little different.
Suppose that $M$ is a module over a commutative ring $R$ with identity and that there $\exists x' \in M^*$ a linear form and $\exists x \in M$ such that $\langle x , x' \rangle$ is invertible.
- How do we show that an element which is not a left divisor of zero in $End_R (M)$ is an injective endomorphism?
(Actually the notes say in parituclar if $M$ is free but I am even confused about this statement.. All we need to show is 1. right and the free case follows?)