We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of...We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn（R） is weakly nil-clean, and to show that the endomorphism ring EndD（V） over a vector space VD is weakly nil-clean if and only if it is nil-clean or dim（V） = 1 with D Z3.展开更多
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...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.展开更多
A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only...A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ∈ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM（R） is 5-potent and BM（R） is nil, if and only if J（R） is nil and R/J（R） is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn（R; s） is Zhou nil-clean if and only if R is Zhou nil-clean and s ∈ J（R）.展开更多
Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpo...Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpotent matrix N.展开更多
文摘We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn（R） is weakly nil-clean, and to show that the endomorphism ring EndD（V） over a vector space VD is weakly nil-clean if and only if it is nil-clean or dim（V） = 1 with D Z3.
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).
文摘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.
基金The authors are grateful to the referee for his/her careful the paper, and for the invaluable comments which improve our presentation reading of author H.Y. Chen was supported by the Natural Science Foundation of Zhejiang （No. LY17A010018）, China. The first Province
文摘A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ∈ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM（R） is 5-potent and BM（R） is nil, if and only if J（R） is nil and R/J（R） is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn（R; s） is Zhou nil-clean if and only if R is Zhou nil-clean and s ∈ J（R）.
基金supported by Ministry of Educations,Science and Technological Development of Republic of Serbia Project#174032.
文摘Let R be an associative unital ring and not necessarily commutative.We analyze conditions under which every n×n matrix A over R is expressible as a sum A=E1+…+Es+N of(commuting)idempotent matrices Ei and a nilpotent matrix N.
基金supported by NSFC(No.11471108)Natural Science Foundation of Hunan Province(Nos.2015JJ2051,2016JJ2050)+1 种基金Scientific Research Foundation of Hunan Provincial Education Department(No.12B101)the Teaching Reform Foundation of Hunan Province(No.G21316)
基金Supported in part by NSFC(No.11401009)Anhui Provincial Natural Science Foundation(No.1408085QA01)the Key Natural Science Foundation of Anhui Educational Committee(No.KJ2014A082)
