If $S$ is a ring and $R$ is a Noetherian subring of $S$, and we know $_RS$ (i.e. $S$ viewed as a left $R$-module) is finitely generated (hence Noetherian), is $_SS$ necessarily Noetherian?
I can't figure it out... it feels like it should be true since $_SS$ contains $_RS$ and we have only finitely more things to worry about.