DefinitionDefinition: 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.

Quotient module $M/N$ with scalar multiplication on additive cosets

Definition

Let $N\le M$ be a submodule of a left $R$ -module. Since the additive group of $M$ is abelian, $N$ is normal, so the additive quotient group $M/N$ consists of the cosets $m+N$ . Its proposed scalar action is

$r(m+N):=rm+N.$

The well-definedness and module laws are established in The quotient action is well defined and makes $M/N$ a module ↗. The resulting module is the quotient module $M/N$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 12 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)

Quotient module M/NM/NM/N with scalar multiplication on additive cosets