Alphabeta MathHow statement and proof provenance work
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A semisimple ring as a ring whose left regular module is semisimple
Definition
A unital ring is semisimple when its left regular module is semisimple. This is a left-module definition and uses no Jacobson radical. For the zero ring, the regular module is zero and hence semisimple; the Wedderburn-Artin theorem below is stated for nonzero rings.
Depends on
Used by
- If char k ∤ |G|, then k[G] is a semisimple ring Corollary
- If char k ∣ |G|, then k[G] is not semisimple Corollary
- Left and right semisimplicity of a ring agree Corollary
- Semisimple rings are left and right Noetherian and Artinian Corollary
- Semilinear Galois actions, twists, and split central idempotents Definition
- Projective representations and twisted algebra modules Lemma
- Equivalent module-theoretic characterizations of semisimple rings Theorem
- For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple Theorem
- Galois orbits classify simple modules after splitting base change Theorem
- Matrix rings over division rings are semisimple Theorem
- Wedderburn–Artin theorem for semisimple rings Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)