Euclidean domain

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, more specifically in <a href="/facts/Ring_theory/VW6svEFe">ring theory</a>, a Euclidean domain (also called a Euclidean ring) is an <a href="/facts/Integral_domain/cylt6zxW">integral domain</a> that can be endowed with a Euclidean function which allows a suitable generalization of <a href="/facts/Euclidean_division/hgNk37et">Euclidean division</a> of <a href="/facts/Integer/PUwwrawx">integers</a>. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the <a href="/facts/Greatest_common_divisor/kNFvBuKl">greatest common divisor</a> 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 (<a href="/facts/B%C3%A9zout%27s_identity/BKYSGV24">Bézout's identity</a>). In particular, the existence of efficient algorithms for Euclidean division of integers and of <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> in one variable over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> is of basic importance in <a href="/facts/Computer_algebra/hjun1KAX">computer algebra</a>.
It is important to compare the <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of Euclidean domains with the larger class of <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a> (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 <a href="/facts/Euclidean_algorithm/wApql29c">Euclidean algorithm</a> and <a href="/facts/Extended_Euclidean_algorithm/wmomforl">extended Euclidean algorithm</a> to compute greatest common divisors. 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.
Every <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> in a Euclidean domain is <a href="/facts/Principal_ideal/Cga66qgM">principal</a>, which implies a suitable generalization of the <a href="/facts/Fundamental_theorem_of_arithmetic/EzHUwlXX">fundamental theorem of arithmetic</a>: every Euclidean domain is also a <a href="/facts/Unique_factorization_domain/aYsX2swC">unique factorization domain</a>. Euclidean domains appear in the following chain of <a href="/facts/Subclass_(set_theory)/FjTF1nAr">class inclusions</a>:

<a href="/facts/Rng_(algebra)/SquNCXKe">rngs</a> ⊃ <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> ⊃ <a href="/facts/Commutative_ring/QS1r2ovR">commutative rings</a> ⊃ <a href="/facts/Integral_domain/cylt6zxW">integral domains</a> ⊃ <a href="/facts/Integrally_closed_domain/L3kgG2q6">integrally closed domains</a> ⊃ <a href="/facts/GCD_domain/PWrf9Tkm">GCD domains</a> ⊃ <a href="/facts/Unique_factorization_domain/aYsX2swC">unique factorization domains</a> ⊃ <a href="/facts/Principal_ideal_domain/JCwAUnyW">principal ideal domains</a> ⊃ euclidean domains ⊃ <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a> ⊃ <a href="/facts/Algebraically_closed_field/HuVQClok">algebraically closed fields</a>

Euclidean domain open-in-new

Euclidean domain