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.
- 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.
Unital left and right modules over a ring; unqualified module means left module
Definition
Let be a ring. A left -module is an abelian group with a scalar action , , satisfying
A right -module has an action , , with the analogous right-handed axioms. Unless “right” is stated, module means a unital left module.
Depends on
Used by
- Annihilators, torsion elements and the torsion subset of a module Definition
- Module homomorphism and isomorphism, kernel, image and cokernel Definition
- Quotient module M/N with scalar multiplication on additive cosets Definition
- Simple module: a nonzero module with no proper nonzero submodule Definition
- Submodule of a module Definition
- Left ideals are exactly the submodules of the regular left module _RR Example
- The residue classes ℤ/n as a ℤ-module Example
- In a module, 0_Rm=0_M, r0_M=0_M, (-r)m=-(rm) and r(-m)=-(rm) Lemma
- The one-step submodule criterion; intersections and sums of submodules are submodules Lemma
- Left modules over a fixed ring and module homomorphisms form the large locally small category R-Mod 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: 10 results over 10 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)