Editing Euclidean domain (section)

{{short description|Commutative ring with a Euclidean division}}
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 [[#Definition|Euclidean function]] which allows a suitable generalization of [[Euclidean division]] of [[integer]]s.  This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the [[ring (mathematics)|ring]] of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the [[Greatest common divisor#In commutative rings|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]]).  In particular, the existence of efficient algorithms for Euclidean division of integers and of [[polynomial]]s in one variable over a [[field (mathematics)|field]] is of basic importance in [[computer algebra]].

It is important to compare the [[class (set theory)|class]] of Euclidean domains with the larger class of [[principal ideal domain]]s (PIDs).  An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but lacks an analogue of the [[Euclidean algorithm]] and [[extended Euclidean algorithm]] to compute greatest common divisors. So, given an integral domain {{mvar|R}}, it is often very useful to know that {{mvar|R}} has a Euclidean function: in particular, this implies that {{mvar|R}} is a PID.  However, if there is no "obvious" Euclidean function, then determining whether {{mvar|R}} is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

Every [[ideal (ring theory)|ideal]] in a Euclidean domain is [[principal ideal|principal]], which implies a suitable generalization of the [[fundamental theorem of arithmetic]]: every Euclidean domain is also a [[unique factorization domain]]. Euclidean domains appear in the following chain of [[subclass (set theory)|class inclusions]]:
{{Commutative ring classes}}
{{Algebraic structures |Ring}}