翻訳と辞書 Words near each other ・ Singular control ・ Singular distribution ・ Singular function ・ Singular homology ・ Singular integral ・ Singular integral operators of convolution type ・ Singular integral operators on closed curves ・ Singular isothermal sphere profile ・ Singular measure ・ Singular perturbation ・ Singular point of a curve ・ Singular point of an algebraic variety ・ Singular SA03 ・ Singular solution ・ Singular spectrum analysis ・ Singular submodule ・ Singular term ・ Singular they ・ Singular trace ・ Singular value ・ Singular value decomposition ・ Singulari Nos ・ Singularia ・ Singularis ・ Singularitarianism ・ Singularity ・ Singularity (audio drama) ・ Singularity (Bill DeSmedt novel) ・ Singularity (climate) ・ Singularity (Joe Morris album) 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 Singular submodule ： ウィキペディア英語版 Singular submodule In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) ''R'' module ''M'' has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in ''R''. In set notation it is usually denoted as $\backslash mathcal(M)=\backslash \backslash ,$. For general rings, $\backslash mathcal(M)$ is a good generalization of the torsion submodule tors(''M'') which is most often defined for domains. In the case that ''R'' is a commutative domain, $tors(M)=\backslash mathcal(M)$. If ''R'' is any ring, $\backslash mathcal(R\_R)$ is defined considering ''R'' as a right module, and in this case $\backslash mathcal(R\_R)$ is a twosided ideal of ''R'' called the right singular ideal of ''R''. Similarly the left handed analogue $\backslash mathcal(\_R\; R)$ is defined. It is possible for $\backslash mathcal(R\_R)\backslash neq\backslash mathcal(\_R\; R)$. ==Definitions== Here are several definitions used when studying singular submodule and singular ideals. In the following, ''M'' is an ''R'' module: *''M'' is called a singular module if $\backslash mathcal(M)=M\backslash ,$. *''M'' is called a nonsingular module if $\backslash mathcal(M)=\backslash \backslash ,$. *''R'' is called right nonsingular if $\backslash mathcal(R\_R)=\backslash \backslash ,$. A left nonsingular ring is defined similarly, using the left singular ideal, and it is entirely possible for a ring to be right-but-not-left nonsingular. In rings with unity it is always the case that $\backslash mathcal(R\_R)\backslash subsetneq\; R\backslash ,$, and so "right singular ring" is not usually defined the same way as singular modules are. Some authors have used "singular ring" to mean "has a nonzero singular ideal", however this usage is not consistent with the usage of the adjectives for modules. 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「Singular submodule」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.