Let $R$ be a commutative ring with unity. We know then that any finitely generated $R$-module is Hopfian i.e. any surjective homomorphism from $M$ to $M$ is bijective ...
My question is: Is there any dual to the above statement ? Something like , is every finitely co-generated module co-hopfian (every injective endomorphism is bijective) ? Or some refinement of this ? I already know that Artinian modules are co-Hopfian so I would be wanting a possible refinement of that.
NOTE: In a dual sense, we do have the following result: If $M$ is torsion less i.e. the natural map $M \to M^{**}$ is injective and $M^*$ is Co-Hopfian, then $M$ is Hopfian ... unfortunately I doubt this is of any help towards the question I'm asking ...