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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 Height (ring theory) ： ウィキペディア英語版 Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension has been introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k''(..., ''x''''n'' ) has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. ==Explanation== We say that a chain of prime ideals of the form $\backslash mathfrak\_0\backslash subsetneq\; \backslash mathfrak\_1\backslash subsetneq\; \backslash ldots\; \backslash subsetneq\; \backslash mathfrak\_n$ has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of $R$ to be the supremum of the lengths of all chains of prime ideals in $R$. Given a prime $\backslash mathfrak$ in ''R'', we define the height of $\backslash mathfrak$, written $\backslash operatorname(\backslash mathfrak)$, to be the supremum of the lengths of all chains of prime ideals contained in $\backslash mathfrak$, meaning that $\backslash mathfrak\_0\backslash subsetneq\; \backslash mathfrak\_1\backslash subsetneq\; \backslash ldots\; \backslash subsetneq\; \backslash mathfrak\_n\; =\; \backslash mathfrak$.〔Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989〕 In other words, the height of $\backslash mathfrak$ is the Krull dimension of the localization of ''R'' at $\backslash mathfrak$. A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. In a Noetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension.〔Eisenbud, D. ''Commutative Algebra'' (1995). Springer, Berlin. Exercise 9.6.〕 A ring is called catenary if any inclusion $\backslash mathfrak\backslash subset\; \backslash mathfrak$ of prime ideals can be extended to a maximal chain of prime ideals between $\backslash mathfrak$ and $\backslash mathfrak$, and any two maximal chains between $\backslash mathfrak$ and $\backslash mathfrak$ have the same length. A ring is called universally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.〔Matsumura, H. ''Commutative Algebra'' (1970). Benjamin, New York. Example 14.E.〕 In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by ''n'' elements is no greater than ''n''. More generally, the height of an ideal I is the infimum of the heights of all prime ideals containing I. In the language of algebraic geometry, this is the codimension of the subvariety of Spec($R$) corresponding to I.〔Matsumura, Hideyuki: "Commutative Ring Theory", page 30–31, 1989〕 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「Krull dimension」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.