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Let $R$ be a ring with unity. Let $M$ be a unitary right $R$-module. It's known that a simple right module over a commutative Noetherian ring has an Artinian injective hull. I wonder what are the possible alternatives of simplicity so that the injective hull $E(M)$ is Artinian. In other words, what are the conditions that can be given to a non-simple right module which guarantees that $E(M)$ is Artinian.

Thanks in advance.

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I see that you have edited the question so that $R$ is commutative, so I've edited this answer. If $R$ is commutative, what you say is true. But there exists a left and right Artinian ring $R$ (therefore right Noetherian) which does not have this property. See Exercise 24.9 of Anderson and Fuller, "Rings and categories of modules" and the remarks in Section 2 of Hirano, "On injective hulls of simple modules", as well as Jans, "On co-Noetherian rings". Rings with the property that you want are called co-Noetherian. Over a co-Noetherian ring, $E(M)$ is Artinian if and only if $M$ is Artinian.

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