Suppose that $R$ is a local Noetherian ring. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is Artinian?
It is easy if $R$ is Cohen-Macaulay, because we know that if there exists a nonzero finitely generated module $M$ of finite injective dimension then $\mathrm{id}\;M=\mathrm{depth}\;R$. So in our case we get $\mathrm{depth}\;R=0$, so if the ring is Cohen-Macaulay we can deduce that it is Artinian. But this should be true in general, any idea of how to prove it?
(I was told that it can be proved with Matlis duality, but since this is an exercise in Bruns-Herzog that comes before Matlis duality there should be a way to prove it without using it)