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.