Ring theory MOC

Semisimple ring

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

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

Proof of equivalence

proof Right iff left follows from Wedderburn–Artin theorem.

---

develop | en | SemBr

Footnotes

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