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Let $M$ be a module over a commutative ring (with unity) $R$. Let $E_R(M)$ denote the injective Hull of $M$ .

If $M$ is torsion-less (i.e. $\cap_{f\in M^*=Hom_R(M,R)} \ker f=(0)$ ) and $E_R(M)$ is Hopfian (i.e. every surjective $R$-module homomorphism $E_R(M) \to E_R(M)$ is injective), then is $M$ Hopfian ? If this is not true in general, what if we assume some more condition on $R$ like Noetherian, local ?

If $R$ is a commutative integral domain, and $M$ is torsion-free, and $E_R(M)$ is Hopfian, then I can show that $M$ is Hopfian. Since torsion-less is even stronger than torsion-free, hence the question for commutative rings which are not necessarily integral domain.

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