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.

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