An integral domain $(R,\cdot,+)$ is called an **Euclidean ring**, if there is a function, called the **Euclidean function**, mapping its non-zero elements to the set $\mathbb N$ of natural numbers $f:R\setminus\{0\}\to\mathbb N$ such that for any two elements $a,b\in R$ we have $$a=qb+r$$

with either $r=0$ ($0\in R$) or $f( r) < f(b).$

- This is a generalization of the division with quotient and remainder of integers.

| | | | | created: 2019-06-29 09:44:07 | modified: 2019-06-29 09:51:06 | by: *bookofproofs* | references: [677], [8250]

(none)

[8250] **Koch, H.; Pieper, H.**: “Zahlentheorie – Ausgewählte Methoden und Ergebnisse”, Studienbücherei, 1976

[677] **Modler, Florian; Kreh, Martin**: “Tutorium Algebra”, Springer Spektrum, 2013

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