I want to prove this :
Over an artinian ring every nonzero module has a simple submodule.
But the same statement for Noetherian rings is not true.
Is there any hint how to show that?
Thank you very much.
2026-03-25 10:59:52.1774436392
Over an artinian ring every nonzero module has a simple submodule?
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Without loss of generality, we can assume that the module is finitely generated (just pick a finitely generated submodule).
A finitely generated module over an Artinian ring is itself Artinian. In particular, every descending chain of submodules stabilises.
So if you didn't have a simple submodule, every submodule has a proper non-trival submodule, leading to an infinite strictly descending chain. This would contradict the module being Artinian.