Hi the following problem is related to the post Example of a an endomorphism which is not a right divisor of zero and not onto.
Below is one of the many facts I am having trouble dealing with in a multi part example centered around standard notions about endomorphism modules of commutative rings with identity. I think the original source for these type of problems comes from one of the Bourbaki books on Algebra.
Suppose $M$ is a left $R$-module such that for every submodule $N \neq 0$ with $N \subset M$ there exits an endomorphsim $f \neq 0$ with $f(M) \subset N$. How do you show an element which is not a left divisor of zero in $End_R(M)$ is injective.