翻訳と辞書 Words near each other ・ Semi-formal ・ Semi-Fowler's position ・ Semi-generic ・ Semi-Gloss ・ Semi-Gloss (album) ・ Semi-Gods Ceiling ・ Semi-Hilbert space ・ Semi-HotRuudolf Megamixtape ・ Semi-implicit Euler method ・ Semi-infinite ・ Semi-infinite programming ・ Semi-inner-product ・ Semi-invariant of a quiver ・ Semi-Italian Opening ・ Semi-Lagrangian scheme ・ Semi-local ring ・ Semi-locally simply connected ・ Semi-log plot ・ Semi-major axis ・ Semi-membership ・ Semi-Mental ・ Semi-minor axis ・ Semi-mobile ・ Semi-Monde ・ Semi-monocoque ・ Semi-On ・ Semi-Onsen Station ・ Semi-Open Game ・ Semi-opera ・ Semi-orthogonal matrix 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 Semi-local ring ： ウィキペディア英語版 Semi-local ring In mathematics, a semi-local ring is a ring for which ''R''/J(''R'') is a semisimple ring, where J(''R'') is the Jacobson radical of ''R''. The above definition is satisfied if ''R'' has a finite number of maximal right ideals (and finite number of maximal left ideals). When ''R'' is a commutative ring, the converse implication is also true, and so the definition of semi-local for commutative rings is often taken to be "having finitely many maximal ideals". Some literature refers to a commutative semi-local ring in general as a ''quasi-semi-local ring'', using semi-local ring to refer to a Noetherian ring with finitely many maximal ideals. A semi-local ring is thus more general than a local ring, which has only one maximal (right/left/two-sided) ideal. == Examples == * Any right or left Artinian ring, any serial ring, and any semiperfect ring is semi-local. * The quotient $\backslash mathbb/m\backslash mathbb$ is a semi-local ring. In particular, if $m$ is a prime power, then $\backslash mathbb/m\backslash mathbb$ is a local ring. * A finite direct sum of fields $\backslash bigoplus\_^n$ is a semi-local ring. * In the case of commutative rings with unity, this example is prototypical in the following sense: the Chinese remainder theorem shows that for a semi-local commutative ring ''R'' with unit and maximal ideals ''m1, ..., mn'' :$R/\backslash bigcap\_^n\; m\_i\backslash cong\backslash bigoplus\_^n\; R/m\_i\backslash ,$. :(The map is the natural projection). The right hand side is a direct sum of fields. Here we note that ∩i mi=J(''R''), and we see that ''R''/J(''R'') is indeed a semisimple ring. * The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. * The endomorphism ring of an Artinian module is a semilocal ring. * Semi-local rings occur for example in algebraic geometry when a (commutative) ring ''R'' is localized with respect to the multiplicatively closed subset ''S = ∩ (R \ pi)'', where the ''pi'' are finitely many prime ideals. 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「Semi-local 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.