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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Semisimple modules as direct sums of simple modules
Definition
A left -module is semisimple when it is an internal direct sum of simple submodules, allowing the empty direct sum. Hence the zero module is semisimple.
Depends on
Used by
- A completely reducible representation as a finite direct sum of irreducible subrepresentations Definition
- A semisimple ring as a ring whose left regular module is semisimple Definition
- Projective representations and twisted algebra modules Lemma
- A finitely generated semisimple module is a finite direct sum of simple modules Theorem
- Equivalent characterizations of semisimple modules Theorem
- Galois orbits classify simple modules after splitting base change Theorem
- Indecomposable projective kG-modules correspond to simple modules through taking the head Theorem
Dependency tree · two levels
4 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)