The following question is related to the following problem which I have been stuck on.
Suppose $M$ is a free $\mathbb{Z}$-module.
If for every submodule $N \neq M$ with $N \subset M$ there exits a linear form $x^{*} \in M^{*}$ which is zero on $N$ and surjective, every element of $End_{R}(M)$ which is not a right divisor of zero is a surjective endomorphism.
Does there exist $\mathbb{Z}$-module homomorphisms from $M \rightarrow M$ that are not surjective and are not a right divisors of zero?