Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-03
How 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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Simple module: a nonzero module with no proper nonzero submodule
Definition
A left -module is simple if and its only submodules are and . Equivalently, has no proper nonzero submodule.
Depends on
Used by
- Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules Corollary
- Composition series and length of a module Definition
- Semisimple modules as direct sums of simple modules Definition
- The radical, socle, head, and Loewy series of a finite-dimensional module Definition
- The socle as the sum of all simple submodules Definition
- A normal p-subgroup acts trivially on every simple module in characteristic p Proposition
- For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple Theorem
- For a finite-dimensional algebra, the module radical is exactly the action of the Jacobson radical Theorem
- Matrix rings over division rings are semisimple Theorem
- Schur's lemma for simple modules Theorem
Dependency tree · two levels
5 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
- McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)