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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?

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up vote 3 down vote accepted

Yes. For example, let $M$ be $\mathbb{Z}$ and consider the endomorphism $n \mapsto 2n$.

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