Here's a quick question on noetherian rings. I know that for a ring $R$, the following are equivalent.
- $R$ is left noetherian
- Every finitely generated left $R$-module is noetherian
- Every submodule of a finitely generated left $R$-module is finitely generated.
Is there a corresponding result with 'left' replaced by 'right' throughout?
The answer is yes.
All of the proofs would go through with "left" replaced by "right" and with the arguments carried out with right modules instead of left modules. That goes for all theorems of this sort, not just the one you mention.