# Euclidean ring

An 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 and hence a 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 can be used to find the greatest common divisor of two elements.

#### References

 [1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
How to Cite This Entry:
Euclidean ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_ring&oldid=33778
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article