Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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.
Submodule of a module
Definition
Let be a left -module. A subset is a submodule when it is a subgroup of the additive group of and is closed under scalars:
The operations on are the restrictions of those of . Write when the ring and module are understood.
Depends on
Used by
- Generated submodule, cyclic and finitely generated modules, module basis and free module Definition
- Quotient module M/N with scalar multiplication on additive cosets Definition
- Simple module: a nonzero module with no proper nonzero submodule Definition
- Left ideals are exactly the submodules of the regular left module _RR Example
- The one-step submodule criterion; intersections and sums of submodules are submodules Lemma
- The canonical map M→ M/N is a surjective module homomorphism with kernel N; thus every submodule is a kernel Proposition
- The quotient action is well defined and makes M/N a module Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 11 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
- McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)