Prove or Disprove: Finitely generated Artinian module is Noetherian.
I think it is true and I am trying to prove it. I am considering reducing the case to Artinian rings. Say $M$ is finitely generated Artinian $R$-module. Then $R/\mathrm{Ann}(M)$ is an Artinian $R$-module. Thus $R/\mathrm{Ann}(M)$ is Artinian as an $R/\mathrm{Ann}(M)$-module. That is, $R/\mathrm{Ann}(M)$ is an Artinian ring. We know that by Hopkins, $R/\mathrm{Ann}(M)$ is a Noetherian ring. Thus $M$ when seen as an $R/\mathrm{Ann}(M)$-module is Noetherian. I wonder how I can prove that $M$ is Noetherian as an $R$-module?
Best Answer
The answer is trivially yes: If $M$ is finitely generated and artinian, then $R/\mathrm{Ann}(M)$ is an artinian ring, hence noetherian. Now just notice that the $R/\mathrm{Ann}(M)$-submodules of $M$ coincide with the $R$-submodules of $M$.