关于Weakly J~#-clean环

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英文篇名：On Weakly J~#-clean Rings 作者：郝亚璞 ; 陈焕艮 英文作者：HAO Yapu;CHEN Huanyin;School of Science,Hangzhou Normal University; 关键词：weakly ; J#-clean环 ; clean环 ; weakly ; nil ; clean环 ; S=R[D ; C]环 ; 幂等元 ; Jacobson根 英文关键词：weakly J#-clean ring;;clean ring;;weakly nil clean;;S = R[D,C] ring;;idempotent;;Jacobson radical 中文刊名：HSFZ 英文刊名：Journal of Hangzhou Normal University(Natural Science Edition) 机构：杭州师范大学理学院; 出版日期：2017-03-13 21:37 出版单位：杭州师范大学学报(自然科学版) 年：2017 期：v.16;No.89 基金：Supported by the Natural Science Foundation of Zhejiang Province(LY17A010018) 语种：英文; 页：HSFZ201702019 页数：9 CN：02 ISSN：33-1348/N 分类号：64-71+77

摘要

一个环R叫做weakly J~#-clean环,如果R中的每一个元素都可以写成a=e+j或a=-e+j的形式,其中e是幂等元,jn属于Jacobson根.在这篇文章中我们证明了R是weakly nil-clean环当且仅当R是weakly J~#-clean环并且J(R)是幂零的.如果I是幂零的,那么R是weakly J~#-clean环当且仅当R/I是weakly J~#-clean环.环R是weakly J~#-clean环当且仅当R/P(R),R×M和幂级数环R[[x]]分别为weakly J~#-clean环.更进一步我们证明以下几点是分别等价的:R是J~#-clean环;存在一个大于等于1的整数n,使得Tn(R)是J~#-clean环;存在一个大于等于2的整数n,使得Tn(R)是weakly J~#-clean环.而且,R是J~#-clean环;存在一个大于等于1的整数n,使得×nR是J~#-clean环;存在一个大于等于2的整数n,使得×nR是weakly J~#-clean环.特殊的,阐述了在某种条件下S=R[D,C]是weakly J~#-clean环.
        A ring R is called a weakly J~#-clean ring if for any a∈R can be written as a = e + j or a =-e + j,in which e is idempotent and jnbelongs to Jacobson radical. This article proves a ring R is a weakly nil-clean ring if and only if R is weakly J~#-clean ring and J( R) is nilpotent. If I is nilpotent,then R is a weakly J~#-clean ring if and only if R/I is a weakly J~#-clean ring. A ring R is a weakly J~#-clean ring if and only if R/P( R),R × M,power series ring R[[x]]are weakly J~#-clean rings respectively. Furthermore,it is proved that the followings are equivalent respectively,R is a J~#-clean ring,there is an integer n≥1 such that Tn( R) is a J~#-clean ring,there is an integer n≥2 such that Tn( R) is a weakly J~#-clean ring. Also,R is a J~#-clean ring,there is an integer n≥1 such that ×nR is a J~#-clean ring,there is an integer n≥2 such that ×nR is a weakly J~#-clean ring. In particular,S = R[D,C] is weakly J~#-clean under certain conditions is exposed.

引文

[1]NICHOLSON W K.Lifting idempotents and exchange rings[J].Transactions of the American Mathematical Socity,1977,229(5):269-278.
    [2]DIESL A J.Nil clean rings[J].Algebra,2013,383(6):197-211.
    [3]BREAZ S,DANCHEV P,ZHOU Y.Rings in which every element is either a sum or a difference of a nilpotent and an idempotent[J].Journal of Algebra and Its Applications,2016,15(8):410-422.
    [4]CHEN H Y.On strongly J-clean rings[J].Comm Algebra,2010,38(10):3790-3804.
    [5]SHEN H D,CHEN H Y.On weakly J-clean rings[J].Journal of Hangzhou Normal University(Natiral Science Edition),2015,14(6):616-624.
    [6]HARTE R E.On quasinilpotents in rings[J].Panamerican Math,1991,1(1):10-16.
    [7]WANG Z,CHEN J L.Pseudo Drazin inverses in associative rings and Banach algebras[J].Linear Algebra Appl,2012,437(6):1332-1345.
    [8]CHENG G P,CHEN J L.The Structure of Ring R[D,C]and Its Characterizations[J].Journal Nanjing University(Natural Sciences Edition),2007,24(1):20-28.