DefinitionDefinition: AI-adaptedProof: Not applicable

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.

Simple module: a nonzero module with no proper nonzero submodule

Definition

A left $R$ -module $M$ is simple if $M \neq 0$ and its only submodules are $0$ and $M$ . Equivalently, $M \neq 0$ and $M$ has no proper nonzero submodule.

Depends on

Dependency tree · 2 levels

Sources

McGerty, Algebra II: Rings and Modules, Section 3