Euclidean ring

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An integral domain with an identity such that to each non-zero element of it corresponds a non-negative integer satisfying the following requirement: For any two elements and with one can find elements and such that

where either or .

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 plays the part of ), and also the ring of polynomials in one variable over a field ( 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.


[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
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Euclidean ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article