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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Euclidean domain ： ウィキペディア英語版 Euclidean domain In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a commutative ring that can be endowed with a Euclidean function (explained 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'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 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: ==Definition== Let ''R'' be an integral domain. A Euclidean function on ''R'' is a function $f$ from $R\; \backslash setminus\; \backslash$ 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 are ''q'' and ''r'' in ''R'' such that and either ''r'' = 0 or . A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. It is important to note that a particular Euclidean function ''f'' is ''not'' part of the structure of a Euclidean domain: in general, a Euclidean domain will admit many different Euclidean functions. Most algebra texts require a Euclidean function to have the following additional property: *(EF2) For all nonzero ''a'' and ''b'' in ''R'', . However, one can show that (EF2) is superfluous in the following sense: any domain ''R'' which can be endowed with a function ''g'' satisfying (EF1) can also be endowed with a function ''f'' satisfying (EF1) and (EF2): indeed, for $\backslash scriptstyle\; a\; \backslash in\; R\; \backslash setminus\; \backslash$ one can define ''f''(''a'') as follows :$f(a)\; =\; \backslash min\_\{x\; \backslash in\; R\; \backslash setminus\; \backslash \{0\backslash \}\}\; g(xa)$ In words, one may define ''f''(''a'') to be the minimum value attained by ''g'' on the set of all non-zero elements of the principal ideal generated by ''a''. A multiplicative Euclidean function is one such that ''f''(''ab'')=''f''(''a'')''f''(''b'') and ''f''(''a'') is never zero. It follows that ''f''(1)=1 and in fact ''f''(''a'')=1 if and only if ''a'' is a unit. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euclidean domain」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.