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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 Ascending chain condition ： ウィキペディア英語版 Ascending chain condition In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings.〔Hazewinkel, Gubareni & Kirichenko (2004), p.6, Prop. 1.1.4.〕〔Fraleigh & Katz (1967), p. 366, Lemma 7.1〕〔Jacobson (2009), p. 142 and 147〕 These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. == Definition == A partially ordered set (poset) ''P'' is said to satisfy the ascending chain condition (ACC) if every strictly ascending sequence of elements eventually terminates. Equivalently, given any sequence :$a\_1\; \backslash ,\backslash leq\backslash ,\; a\_2\; \backslash ,\backslash leq\backslash ,\; a\_3\; \backslash ,\backslash leq\backslash ,\; \backslash cdots,$ there exists a positive integer ''n'' such that :$a\_n\; =\; a\_\; =\; a\_\; =\; \backslash cdots.$ Similarly, ''P'' is said to satisfy the descending chain condition (DCC) if every strictly descending sequence of elements eventually terminates, that is, there is no infinite descending chain. Equivalently every descending sequence :$\backslash cdots\; \backslash ,\backslash leq\backslash ,\; a\_3\; \backslash ,\backslash leq\backslash ,\; a\_2\; \backslash ,\backslash leq\backslash ,\; a\_1$ of elements of ''P'', eventually stabilizes. 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「Ascending chain condition」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.