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