All rings are with unity but not necessarily commutative.

We know that if a ring $R$ is left semisimple (a direct sum of minimal left ideals), then it is left artinian.

My question is: if an unital $R$-module $_R M$ is semisimple (a direct sum of simple $R$-submodule of $_R M$), is $_R M$ left artinian?


No, there is no direct relationship: you can take any simple module $S$ and then $\oplus_{i\in \mathbb N} S$ is semisimple but not Artinian.

Over a field every module is semisimple, but the only Artinian ones are the finitely generated ones.

  • 2
    $\begingroup$ Thanks. You are always like my mentor when I am learning algebra. $\endgroup$ – bfhaha Sep 19 '17 at 12:00
  • $\begingroup$ @bfhaha Glad to be able to help! $\endgroup$ – rschwieb Sep 19 '17 at 12:02

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