Euclidean domain

Definition

A Euclidean domain is an integral domain $R$ with a function $\delta \colon R \setminus \{0\} \to \mathbb{N}$ with this property. For all $a, b \in R$ with $b \neq 0$ , there are $q, r \in R$ such that $a = b q + r$ , and $r = 0$ or $\delta(r) < \delta(b)$ . The function $\delta$ is a Euclidean function of $R$ .

Remarks

The absolute value is a Euclidean function on $\mathbb{Z}$ . The degree is a Euclidean function on $K[x]$ for a field $K$ . Some authors also require $\delta(a) \le \delta(a b)$ for all nonzero $a, b$ . This changes the class of Euclidean domains only in appearance, because a ring with a Euclidean function has one with this extra property. Every Euclidean domain is a principal ideal domain.

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Group concept
Ring definition
Commutative ring definition
Zero divisor definition
Integral domain definition
Euclidean domain definition

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29 pages build on this one.

Tag	`0GJ2HRG95T`
URI	`m2://0GJ2HRG95T`
Kind	definition
Writing	W1 stated
MSC	13F07
Defines	Euclidean domain
Uses	Integral domain

Euclidean domain

Definition

Remarks

Reading path

Where this leads

Further reading

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