Euclidean domain

Euclid 环

Algebra & Algebraic Geometry · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: ED

Translation evidence: Literature usage

Meaning in context

A domain with a function $\delta:R\setminus\{0\}\to\mathbb Z_{\ge0}$ such that $a=bq+r$ with $r=0$ or $\delta(r)<\delta(b)$ whenever $b\ne0$.

具有非零元素上的非负整数函数并能进行使余数函数值严格下降的带余除法的整环。

Example

Polynomial degree makes $F[x]$ Euclidean.

Usage and distinctions

This record specializes Li's well-ordered codomain to $L=\mathbb Z_{\ge0}$, as in his polynomial-degree example; the Euclidean function need not be ordinary absolute value.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Wen-Wei Li — Methods in Algebra I, author source唯一分解性 (sec:UFD), lemma prop:Euclidean-ring, division condition specialized to L=Z_{≥0}; following example eg:polynomial-PID uses this integer-valued codomain · Checked 2026-10-15

Translation sources

1. 李文威 — 代数学方法 卷一，作者源文件唯一分解性 (sec:UFD), lemma prop:Euclidean-ring, final sentence naming Euclid 环; integer-valued example eg:polynomial-PID · Checked 2026-10-15

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