DefinitionDefinition: 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.

Unital left and right modules over a ring; unqualified module means left module

Definition

Let $R$ be a ring. A left $R$ -module is an abelian group $(M, +, 0_{M})$ with a scalar action $R \times M \to M$ , $(r, m) \mapsto r m$ , satisfying

$r (m + n) = r m + r n, (r + s) m = r m + s m, (r s) m = r (s m), 1_{R} m = m .$

A right $R$ -module has an action $M \times R \to M$ , $(m, r) \mapsto m r$ , with the analogous right-handed axioms. Unless “right” is stated, module means a unital left module.

Depends on

Dependency tree · 2 levels

Sources

McGerty, Algebra II: Rings and Modules, Section 3