Ring theory MOC

Semisimple ring

A ring š¯‘… is semisimple iff it meets any of the following equivalent conditions:1 #m/def/ring

  1. š¯‘… is semisimple as a left module;
  2. š¯‘… is semisimple as a right module;
  3. Every š¯‘…-module is semisimple;
  4. Every short exact sequence of š¯‘…-modules splits.
  5. Every š¯‘…-module is projective.
Proof of equivalence

#missing/proof Right iff left follows from Wedderburnā€“Artin theorem.

#state/develop | #lang/en | #SemBr

Footnotes

  1. 2015. Advanced modern algebra, p. 497 ā†©