Associative rings and algebras Semisimple rings and Wedderburn theory Concept
Semisimple ring
Summary
A semisimple ring is a ring that is a semisimple module over itself: it is a direct sum of minimal left ideals. Every module over a semisimple ring is semisimple, so every submodule has a complement and every module is projective and injective. The Artin–Wedderburn theorem describes them completely as finite products of matrix rings over division rings. Group algebras of finite groups over fields of characteristic zero are the main examples, by Maschke's theorem.
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About this page
| Tag | NY25HXA478 |
|---|---|
| URI | m2://NY25HXA478 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 16D60, 16K99, 16P20 |
| Specializes | Ring |
| Changed | 2 November 2026 |
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Used by
- Maschke's theorem
- Artinian reduced rings are semisimple
- Separable algebra
- Goldie's theorem
- Artin–Wedderburn theorem
- Jordan–Zassenhaus theorem
- Hopkins–Levitzki theorem
- Semilocal ring