DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The socle as the sum of all simple submodules

Definition

For a left $R$ -module $M$ , the socle is $Soc (M) : = \sum {S \leq M : S is simple} .$ Concretely it is the submodule generated by the union of all simple submodules. If there are none, the sum is $0$ , so the definition is always meaningful.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 8 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

MIT 18.706, Lecture 2: Semisimple Modules, Socles, Artinian Rings, Wedderburn's Theorem (standard reference, not scraped)