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- Euclidean_domain abstract "In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail 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 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 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: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields".
- Euclidean_domain wikiPageID "10376".
- Euclidean_domain wikiPageRevisionID "605112661".
- Euclidean_domain hasPhotoCollection Euclidean_domain.
- Euclidean_domain subject Category:All_articles_lacking_sources.
- Euclidean_domain subject Category:Commutative_algebra.
- Euclidean_domain subject Category:Euclid.
- Euclidean_domain subject Category:Ring_theory.
- Euclidean_domain comment "In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.".
- Euclidean_domain label "Anneau euclidien".
- Euclidean_domain label "Dominio euclideo".
- Euclidean_domain label "Dominio euclídeo".
- Euclidean_domain label "Domínio euclidiano".
- Euclidean_domain label "Dziedzina Euklidesa".
- Euclidean_domain label "Euclidean domain".
- Euclidean_domain label "Euclidisch domein".
- Euclidean_domain label "Euklidischer Ring".
- Euclidean_domain label "Евклидово кольцо".
- Euclidean_domain label "ユークリッド環".
- Euclidean_domain label "歐幾里得整環".
- Euclidean_domain sameAs Eukleidovský_obor.
- Euclidean_domain sameAs Euklidischer_Ring.
- Euclidean_domain sameAs Ευκλείδεια_περιοχή.
- Euclidean_domain sameAs Dominio_euclídeo.
- Euclidean_domain sameAs Anneau_euclidien.
- Euclidean_domain sameAs Dominio_euclideo.
- Euclidean_domain sameAs ユークリッド環.
- Euclidean_domain sameAs 유클리드_정역.
- Euclidean_domain sameAs Euclidisch_domein.
- Euclidean_domain sameAs Dziedzina_Euklidesa.
- Euclidean_domain sameAs Domínio_euclidiano.
- Euclidean_domain sameAs m.02tb6.
- Euclidean_domain sameAs Q867345.
- Euclidean_domain sameAs Q867345.
- Euclidean_domain wasDerivedFrom Euclidean_domain?oldid=605112661.
- Euclidean_domain isPrimaryTopicOf Euclidean_domain.