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.
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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
Euler characteristic of a finite complex of finite-rank free abelian groups Definition
The free rank of a finitely generated module over a PID Definition
Euler-Poincare formula for finite free complexes Theorem
Every nonzero commutative ring has invariant basis number for finite bases Theorem
Simultaneous bases for a submodule of a finite free module over a PID Theorem

Dependency tree · two levels

7 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

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)