Ring theory MOC

Semisimple ring

A ring is semisimple iff it meets any of the following equivalent conditions:¹ ring

1. is semisimple as a left module;
2. is semisimple as a right module;
3. Every -module is semisimple;
4. Every short exact sequence of -modules splits.

Proof of equivalence

proof Right iff left follows from Wedderburn–Artin theorem.

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develop | en | sembr

Footnotes

1. 2015. Advanced modern algebra, p. 497 ↩