LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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.

For a finite-length module, the radical is a superfluous submodule

Statement

Let $M$ be a finite-length module. If $N \leq M$ and $N + rad (M) = M$ , then $N = M$ .

Facts & Assumptions

Given: A finite-length module $M$ and a submodule $N \leq M$ with $N + rad (M) = M$ .

[F1]

The module radical is the intersection of the maximal submodules, and the head is $M / rad (M)$ (The radical, socle, head, and Loewy series of a finite-dimensional module).

[L1]

Finite length means a composition series exists (Composition series and length of a module).

[L2]

Every nonzero finitely generated module has a maximal proper submodule (Under Choice, every finitely generated nonzero module has a maximal proper submodule).

Proof

technique · direct

1.1

Assume for contradiction that $N \neq M$ . Since $M$ has finite length by [L1], it is finitely generated. Therefore the nonzero quotient $M / N$ has a maximal proper submodule by [L2], and its inverse image in $M$ is a maximal submodule $P$ containing $N$ .

L1L2givenassume-contraalgebra

2.1

By [F1], the radical lies in every maximal submodule, so $rad (M) \subseteq P$ . Hence $M = N + rad (M) \subseteq P < M$ , a contradiction. Therefore $N = M$ , and the radical is superfluous.

F1step 1.1discharge-contradiction∎

Depends on

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)