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Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder

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 $f( r) < f(b).$

Notes

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

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References

Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013
2. Koch, H.; Pieper, H.: "Zahlentheorie - Ausgewählte Methoden und Ergebnisse", Studienbücherei, 1976