Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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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
- Left and right semisimplicity of a ring agree Corollary
- Semisimple rings are left and right Noetherian and Artinian Corollary
- Equivalent module-theoretic characterizations of semisimple rings Theorem
- Matrix rings over division rings are semisimple Theorem
- Wedderburn–Artin theorem for semisimple rings Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 13 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- William Crawley-Boevey, Noncommutative Algebra, Chapter 1 Sections 1.1-1.9 (standard reference, not scraped)