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Concept
Euclidean domain
Summary
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of 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.
It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use the Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.
So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID. However, if there is no "obvious" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.
Euclidean domains appear in the following chain of class inclusions:
Let R be an integral domain. A Euclidean function on R is a function f from R  {0} to the non-negative integers satisfying the following fundamental division-with-remainder property:
(EF1) If a and b are in R and b is nonzero, then there exist q and r in R such that a = bq + r and either r = 0 or f (r) < f (b).
Official source
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