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

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

Definition

Depends on

Dependency tree · 2 levels

Sources