DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-03

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.

Euclidean domain and Euclidean function

Definition

An integral domain $R$ is a Euclidean domain if it has a function

$\delta:R\setminus\{0\}\longrightarrow\mathbb N$

such that for every $a\in R$ and nonzero $b\in R$ , there are $q,r\in R$ with

$a=bq+r,\qquad r=0\ \text{or}\ \delta(r)<\delta(b).$

Such a function $\delta$ is a Euclidean function. No multiplicative monotonicity condition on $\delta$ is included in this convention.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 15 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

Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)