A ring without zero divisors in which an integer norm and an associated division algorithm (i.e., a Euclidean algorithm) can be defined. For signed integers, the usual norm is the absolute value and the division algorithm gives the ordinary quotient and remainder. For polynomials, the norm is the degree. Important examples of Euclidean rings (besides Z) are the Gaussian integers and C[x], the ring of polynomials with complex coefficients. All Euclidean rings are also principal rings.

Euclidean algorithm | principal ring | ring