在抽象代数中，欧几里得整环（Euclidean domain）是一种能作辗转相除法的整环。凡欧几里得整环必为主理想环。

一个欧几里得整环是一整环 及函数 ，使之满足下述性质：

• 若 而 ，则存在 使得 ，而且或者 ，或者 。
• 若 整除 ，则 。

函数 可设想成元素大小的量度，当 时可取 。

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a commutative ring that can be endowed with a Euclidean function (explained below) which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.