DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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 trace of a square matrix over a commutative ring

Definition

Let $R$ be a commutative ring, let $p \in N$ , and let $A = (a_{i j}) \in M_{p} (R)$ (Finite rectangular matrices over a commutative ring, their entries, rows and columns). The trace of $A$ over $R$ is

${tr}_{R} (A) : = \sum_{i < p} a_{i i} .$

For $p = 0$ , this is the empty sum and equals $0_{R}$ . The subscript may be omitted when the coefficient ring is clear.

Depends on

Finite rectangular matrices over a commutative ring, their entries, rows and columns

Used by

Dependency tree · two levels

4 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

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Section 4.7 (standard reference, not scraped)
S. Axler, Linear Algebra Done Right, 4th ed., Definition 8.47 (standard reference, not scraped)