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

Depends on

Used by

A regular module with bases of sizes one and two Counterexample
Every nonzero commutative ring has invariant basis number for finite bases Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 7 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

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)