DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

Composition series and length of a module

Definition

A composition series of a left $R$ -module $M$ is a finite chain $0 = M_{0} < M_{1} < \dots < M_{n} = M$ whose factors $M_{i} / M_{i - 1}$ are simple. If such a series exists, the length $ℓ_{R} (M)$ is its number $n$ of factors; Jordan–Hölder theorem for modules ↗ proves independence of the chosen series. The zero module has the empty series and length $0$ .

Depends on

Dependency tree · 2 levels

Sources

Arvind Nair, Algebra I, Lecture 5