DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

Upper triangular, lower triangular and diagonal square matrices over a commutative ring

Definition

Let $R$ be a commutative ring, let $n \geq 1$ , and let $A = (a_{i j}) \in M_{n} (R)$ . It is upper triangular when $a_{i j} = 0$ for $i > j$ , lower triangular when $a_{i j} = 0$ for $i < j$ , and diagonal when $a_{i j} = 0$ whenever $i \neq j$ .

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

S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)
D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)