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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 Quasiregular element ： ウィキペディア英語版 Quasiregular element : ''This article addresses the notion of quasiregularity in the context of ring theory, a branch of modern algebra. For other notions of quasiregularity in mathematics, see the disambiguation page quasiregular.'' In mathematics, specifically ring theory, the notion of quasiregularity provides a computationally convenient way to work with the Jacobson radical of a ring.〔Isaacs, p. 180〕 Intuitively, quasiregularity captures what it means for an element of a ring to be "bad"; that is, have undesirable properties.〔Isaacs, p. 179〕 Although a "bad element" is necessarily quasiregular, quasiregular elements need not be "bad," in a rather vague sense. In this article, we primarily concern ourselves with the notion of quasiregularity for unital rings. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory. ==Definition== Let ''R'' be a ring (with unity) and let ''r'' be an element of ''R''. Then ''r'' is said to be quasiregular, if 1 − ''r'' is a unit in ''R''; that is, invertible under multiplication.〔 The notions of right or left quasiregularity correspond to the situations where 1 − ''r'' has a right or left inverse, respectively.〔 An element ''x'' of a non-unital ring is said to be right quasiregular if there is ''y'' such that $x\; +\; y\; -\; xy\; =\; 0$.〔Lam, Ex. 4.2, p. 50〕 The notion of a left quasiregular element is defined in an analogous manner. The element ''y'' is sometimes referred to as a right quasi-inverse of ''x''.〔Polcino & Sehgal (2002), (p. 298 ).〕 If the ring is unital, this definition quasiregularity coincides with that given above.〔Lam, Ex. 4.2(3), p. 50〕 If one writes $x\; \backslash cdot\; y\; =\; x\; +\; y\; -\; xy$, then this binary operation $\backslash cdot$ is associative.〔Lam, Ex. 4.1, p. 50〕 In fact, the map $(R,\backslash cdot)\backslash to(R,\backslash times);\; x\backslash mapsto1-x$ (where × denotes the multiplication of the ring ''R'') is a monoid isomorphism.〔 Therefore, if an element possesses both a left and right quasi-inverse, they are equal.〔Since ''0'' is the multiplicative identity, if $x\; \backslash cdot\; y\; =\; 0\; =\; y\text{'}\; \backslash cdot\; x$, then $y\; =\; (y\text{'}\; \backslash cdot\; x)\; \backslash cdot\; y\; =\; y\text{'}\; \backslash cdot\; (x\; \backslash cdot\; y)\; =\; y\text{'}$. Quasiregularity does not require the ring to have a multiplicative identity. 〕 Note that some authors use different definitions. They call an element ''x'' right quasiregular if there exists ''y'' such that $x\; +\; y\; +\; xy\; =\; 0$,〔Kaplansky, p. 85〕 which is equivalent to saying that 1 + ''x'' has a right inverse when the ring is unital. If we write $x\backslash circ\; y=x+y+xy$, then $(-x)\backslash circ(-y)=-(x\backslash cdot\; y)$, so we can easily go from one set-up to the other by changing signs.〔Lam, p. 51〕 For example, ''x'' is right quasiregular in one set-up iff −''x'' is right quasiregular in the other set-up.〔 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「Quasiregular element」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.