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Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the category of Noetherian and Artinian $R$-modules.

My question is : How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules ? Under what suitable assumptions on $M$, can we conclude that $M$ is Hopfian (resp. Co-hopfian) if (or only if) $D(M)$ is Co-hopfian (resp. Hopfian) ?

NOTE: Hopfian means every surjective endomorphism is injective. Co-hopfian means every injective endomorphism is surjective

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If $D(M)$ is Hopfian (Co-Hopfian), then $M$ is Co-Hopfian (Hopfian resp.). Let me prove one of these, the other being similar. So, assume that $D(M)$ is Hopfian and let $f:M\to M$ be injective. Then, we have an exact sequence, $0\to D(M/f(M))\to D(M)\to D(M)\to 0$, and thus by assumption, $D(M/f(M))=0$. Since $E$ is the injective hull of $k$, easy to see that $M/f(M)=0$.

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