翻訳と辞書 Words near each other ・ Vong v. Sansom ・ Vongchanh Phommavan ・ Vongchavalitkul University ・ Vonges ・ Vongkithem ・ Vongnes ・ Vongo ・ Vongoda River ・ Vongole Fisarmonica ・ Vongsa ・ Von Neumann entropy ・ Von Neumann machine ・ Von Neumann neighborhood ・ Von Neumann paradox ・ Von Neumann programming languages ・ Von Neumann regular ring ・ Von Neumann stability analysis ・ Von Neumann universal constructor ・ Von Neumann universe ・ Von Neumann's inequality ・ Von Neumann's theorem ・ Von Neumann–Bernays–Gödel set theory ・ Von Neumann–Morgenstern utility theorem ・ Von Neumann–Wigner interpretation ・ Von Null auf 42 ・ Von nun ab, Herr Kunze ・ Von Ogden Vogt ・ Von Ormy, Texas ・ Von Otter ・ Von Otterøya Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Von Neumann regular ring ： ウィキペディア英語版 Von Neumann regular ring In mathematics, a von Neumann regular ring is a ring ''R'' such that for every ''a'' in ''R'' there exists an ''x'' in ''R'' such that . To avoid the possible confusion with the regular rings and regular local rings of commutative algebra (which are unrelated notions), von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left module is flat. One may think of ''x'' as a "weak inverse" of ''a''. In general ''x'' is not uniquely determined by ''a''. Von Neumann regular rings were introduced by under the name of "regular rings", during his study of von Neumann algebras and continuous geometry. An element ''a'' of a ring is called a von Neumann regular element if there exists an ''x'' such that .〔Kaplansky (1972) p.110〕 An ideal $\backslash mathfrak$ is called a (von Neumann) regular ideal if it is a von Neumann regular non-unital ring, i.e. if for every element ''a'' in $\backslash mathfrak$ there exists an element ''x'' in $\backslash mathfrak$ such that .〔Kaplansky (1972) p.112〕 == Examples == Every field (and every skew field) is von Neumann regular: for we can take .〔 An integral domain is von Neumann regular if and only if it is a field. Another example of a von Neumann regular ring is the ring M''n''(''K'') of ''n''-by-''n'' square matrices with entries from some field ''K''. If ''r'' is the rank of , then there exist invertible matrices ''U'' and ''V'' such that : 0 &0\end V (where ''I''''r'' is the ''r''-by-''r'' identity matrix). If we set , then : 0 &0\end \beginI_r &0\\ 0 &0\end V = U \beginI_r &0\\ 0 &0\end V = A. More generally, the matrix ring over a von Neumann regular ring is again a von Neumann regular ring.〔 The ring of affiliated operators of a finite von Neumann algebra is von Neumann regular. A Boolean ring is a ring in which every element satisfies . Every Boolean ring is von Neumann regular. 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「Von Neumann regular ring」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.