View source for Euclidean ring

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An [[Integral domain|integral domain]] with an identity such that to each non-zero element $a$ of it corresponds a non-negative integer $n(a)$ satisfying the following requirement: For any two elements $a$ and $b$ with $b\neq0$ one can find elements $q$ and $r$ such that

$$a=bq+r,$$

where either $r=0$ or $n(r)<n(b)$.

Every Euclidean ring is a [[Principal ideal ring|principal ideal ring]] and hence a [[Factorial ring|factorial ring]]; however, there exist principal ideal rings that are not Euclidean. Euclidean rings include the ring of integers (the absolute value $|a|$ plays the part of $n(a)$), and also the ring of polynomials in one variable over a field ($n(a)$ is the degree of the polynomial). In any Euclidean ring the [[Euclidean algorithm|Euclidean algorithm]] can be used to find the [[greatest common divisor]] of two elements.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>