DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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.

Invariant basis number and the rank of a free module

Definition

A unital ring $R$ has invariant basis number for finite bases if $R^{m} ≅ R^{n} ⟹ m = n$ for all $m, n \in N$ , as left $R$ -modules (The free module on a set and its standard basis, Module homomorphism and isomorphism, kernel, image and cokernel).

When $R$ has this property and a free module $F$ has a finite basis of $n$ elements, its rank is ${rank}_{R} (F) = n$ . This definition makes no assertion about equality of arbitrary infinite bases.

Invariant basis number and the rank of a free module

Definition

Depends on

Dependency tree · 2 levels

Sources