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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 Dimension theory (algebra) ： ウィキペディア英語版 Dimension theory (algebra) In mathematics, dimension theory is a branch of commutative algebra studying the notion of the dimension of a commutative ring, and by extension that of a scheme. The theory is much simpler for an affine ring; i.e., an integral domain that is a finitely generated algebra over a field. By Noether's normalization lemma, the Krull dimension of such a ring is the transcendence degree over the base field and the theory runs in parallel with the counterpart in algebraic geometry; cf. Dimension of an algebraic variety. The general theory tends to be less geometrical; in particular, very little works/is known for non-noetherian rings. (Kaplansky's commutative rings gives a good account of the non-noetherian case.) Today, a standard approach is essentially that of Bourbaki and EGA, which makes essential use of graded modules and, among other things, emphasizes the role of multiplicities, the generalization of the degree of a projective variety. In this approach, Krull's principal ideal theorem appears as a corollary. Throughout the article, $\backslash operatorname$ denotes Krull dimension of a ring and $\backslash operatorname$ the height of a prime ideal (i.e., the Krull dimension of the localization at that prime ideal.) Rings are assumed to be commutative except in the last section on dimensions of non-commutative rings. == Basic results == Let ''R'' be a noetherian ring or valuation ring. Then :$\backslash operatorname\; R()\; =\; \backslash operatorname\; R\; +\; 1.$ If ''R'' is noetherian, this follows from the fundamental theorem below (in particular, Krull's principal ideal theorem.) But it is also a consequence of the more precise result. For any prime ideal $\backslash mathfrak$ in ''R'', :$\backslash operatorname(\backslash mathfrak\; R())\; =\; \backslash operatorname(\backslash mathfrak)$. :$\backslash operatorname(\backslash mathfrak)\; =\; \backslash operatorname(\backslash mathfrak)\; +\; 1$ for any prime ideal $\backslash mathfrak\; \backslash supsetneq\; \backslash mathfrak\; R()$ in $R()$ that contracts to $\backslash mathfrak$. This can be shown within basic ring theory (cf. Kaplansky, commutative rings). By the way, it says in particular that in each fiber of $\backslash operatorname\; R()\; \backslash to\; \backslash operatorname\; R$, one cannot have a chain of primes ideals of length $\backslash ge\; 2$. Since an artinian ring (e.g., a field) has dimension zero, by induction, one gets the formula: for an artinian ring ''R'', :$\backslash operatorname\; R(\backslash dots,\; x\_n)\; =\; n.$ 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「Dimension theory (algebra)」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.