# 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://www.encyclopediaofmath.org/index.php?title=Euclidean_ring&oldid=33778