In this paper,we study reduced rings in which every element is a sum of three tripotents that commute,and determine the integral domains over which every n£n matrix is a sum of three tripotents.It is proved that ...In this paper,we study reduced rings in which every element is a sum of three tripotents that commute,and determine the integral domains over which every n£n matrix is a sum of three tripotents.It is proved that for an integral domain R,every matrix in M_(n)(R)is a sum of three tripotents if and only if R■Zp with p=2,3,5 or 7.展开更多
Suppose R is a commutative ring with 1, and 2 is a unit of R. Let Tn(R) be the n × n upper triangular matrix modular over R, and let (?)i(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that prese...Suppose R is a commutative ring with 1, and 2 is a unit of R. Let Tn(R) be the n × n upper triangular matrix modular over R, and let (?)i(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that preserve involutory or tripotent. The main result in this paper is that f ∈ (?)i(R) if and only if there exists an invertible matrix U ∈ Tn(R) and orthogonal idempotent elements e1,e2,e3 ande4 in R with such that where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
基金Supported by Key Laboratory of Financial Mathematics of Fujian Province University(Putian University)(JR202203)the NSF of Anhui Province(2008085MA06).
文摘In this paper,we study reduced rings in which every element is a sum of three tripotents that commute,and determine the integral domains over which every n£n matrix is a sum of three tripotents.It is proved that for an integral domain R,every matrix in M_(n)(R)is a sum of three tripotents if and only if R■Zp with p=2,3,5 or 7.
基金Foundation item:The NSF(10271021)of China and NSF(10531130)of Heilongjiang Education Committee
文摘Suppose R is a commutative ring with 1, and 2 is a unit of R. Let Tn(R) be the n × n upper triangular matrix modular over R, and let (?)i(R) (i=2 or 3) be the set of all R-module automorphisms on Tn(R) that preserve involutory or tripotent. The main result in this paper is that f ∈ (?)i(R) if and only if there exists an invertible matrix U ∈ Tn(R) and orthogonal idempotent elements e1,e2,e3 ande4 in R with such that where
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）.
