Nil-Clean Rings with Involution  

作　　者：Jian Cui Guoli Xia Yiqiang Zhou 

机构地区：[1]Department of Mathematics,Anhui Normal University,Wuhu,Anhui 241002,China [2]Department of Mathematics and Statistics,Memorial University of Newfoundland St.John’s,NL A1C 5S7,Canada

出　　处：《Algebra Colloquium》2021年第3期367-378,共12页代数集刊（英文版）

基　　金：This research was supported by Anhui Provincial Natural Science Foundation(No.2008085MA06);the Key Project of Anhui Education Committee(No.gxyqZD2019009)(for Cui);a Discovery Grant from NSERC of Canada(for Xia and Zhou).

摘　　要：A*-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil*-clean rings are the version of nil-clean rings introduced by Diesl.This paper is about the nil*-clean property of rings with emphasis on matrix rings.We show that a*-ring R is nil*-clean if and only if J(R)is nil and R/J(R)is nil*-clean.For a 2-primal*-ring R,with the induced involution given by(aij)*=(a*ij)^(T),the nil*-clean property of Mn(R)is completely reduced to that of Mn(Zn).Consequently,Mn(R)is not a nil*-clean ring for n=3,4,and M2(R)is a nil*-clean ring if and only if J(R)is nil,R/J(R)is a Boolean ring and a*-a∈J(R)for all a∈R.

关 键 词：IDEMPOTENT NILPOTENT PROJECTION nil-clean ring *-ring nil*-clean ring matrix ring 

分 类 号：O15[理学—数学]