A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper ...A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper （triangular） matrix ring is one-sided McCoy. It is shown that for many polynomial extensions, a ring R is right Mccoy if and only if the polynomial extension over R is right Mccoy.展开更多
The Casimir element of a fusion ring （R, B） gives rise to the so called Casimir matrix C of （R,B）. This enables us to construct a generalized Cartan matrix D - C in the sense of Kac for a suitable diagonal matrix ...The Casimir element of a fusion ring （R, B） gives rise to the so called Casimir matrix C of （R,B）. This enables us to construct a generalized Cartan matrix D - C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D - C of finite （resp. affine） type. It turns out that there exists a fusion ring with D - C being of finite （resp. affine） type if and only if D - C has only the form A2 （resp. A1^（1）））. We also realize all fusion rings with D - C being a particular generalized Cartan matrix of indefinite type.展开更多
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
In this paper, two different ring networks with unidirectional couplings and with bidirectional couplings were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two diffe...In this paper, two different ring networks with unidirectional couplings and with bidirectional couplings were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures by cutting a link are completely opposite. The synchronizing ability will decrease if the change is from bidirectional ring to bidirectional chain. Moreover, the change on synchronizing ability will be four times if the number of N is large enough. However, it will increase obviously from unidirectional ring to unidirectional chain. It will be N^2/（π^2） times if the number of N is large enough. The numerical simulations confirm the conclusion in quality. This paper also discusses the effects on synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. Theoretical results are accordant to numerical simulations. Synchronization is an essential physics problem. These results proposed in this paper have some important reference meanings on the real world networks, such as the bioecological system networks, the designing of the circuit, etc.展开更多
We consider the sufficient and necessary conditions for the formal triangular matrix ring being right minsymmetric, right DS, semicommutative, respectively.
A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorph...A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.展开更多
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
Let R be a ring.We show in the paper that the subring Un（R） of the upper triangular matrix ring Tn（R） is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,whe...Let R be a ring.We show in the paper that the subring Un（R） of the upper triangular matrix ring Tn（R） is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,where α is a ring endomorphism of R with α（1） = 1.展开更多
Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group invers...Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices（ABCD）over a ring with unity 1 under the following conditions respectively： （i） B = C, D = 0,B#and（BπA0#both exist; （ii） B is invertible and m = n;（iii）A#and （D - CA#B）# both exist, C = CAA#, where A and D are m × m and n × n matrices, respectively.展开更多
An associative ring with identity R is called Armendariz if, whenever （∑^m i=0^aix^i）（∑^n j=0^bjx^j）=0 in R[x],aibj=0 for all i and j. An associative ring with identity is called reduced if it has no non-zero ni...An associative ring with identity R is called Armendariz if, whenever （∑^m i=0^aix^i）（∑^n j=0^bjx^j）=0 in R[x],aibj=0 for all i and j. An associative ring with identity is called reduced if it has no non-zero nilpotent elements. In this paper, we define a general reduced ring （with or without identity） and a general Armendariz ring （with or without identity）, and identify a class of maximal general Armendariz subrings of matrix rings over general reduced rings.展开更多
Let M be a monoid. Maximal M-Armendariz subrings of upper triangular matrix rings are identified when R is M-Armendariz and reduced. Consequently, new families of M- Armendariz rings are presented.
In this paper we continue the study of various ring theoretic properties of Morita contexts.Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangu...In this paper we continue the study of various ring theoretic properties of Morita contexts.Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch,completely primary,quasi-duo,2-primal,NI,semiprimitive,projective-free,etc.We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective.展开更多
基金The NNSF(10571026)of Chinathe Specialized Research Fund(20060286006)for the Doctoral Program of Higher Education.
文摘A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper （triangular） matrix ring is one-sided McCoy. It is shown that for many polynomial extensions, a ring R is right Mccoy if and only if the polynomial extension over R is right Mccoy.
基金Supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.15KJB110013)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20150537)NSFC(Grant No.11471282)
文摘The Casimir element of a fusion ring （R, B） gives rise to the so called Casimir matrix C of （R,B）. This enables us to construct a generalized Cartan matrix D - C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D - C of finite （resp. affine） type. It turns out that there exists a fusion ring with D - C being of finite （resp. affine） type if and only if D - C has only the form A2 （resp. A1^（1）））. We also realize all fusion rings with D - C being a particular generalized Cartan matrix of indefinite type.
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the National Basic Research 973 Program of China (Grant No. 2006CB708302)the National Natural Science Foundation of China (Grant Nos. 60574045 and 90604005)
文摘In this paper, two different ring networks with unidirectional couplings and with bidirectional couplings were discussed by theoretical analysis. It was found that the effects on synchronizing ability of the two different structures by cutting a link are completely opposite. The synchronizing ability will decrease if the change is from bidirectional ring to bidirectional chain. Moreover, the change on synchronizing ability will be four times if the number of N is large enough. However, it will increase obviously from unidirectional ring to unidirectional chain. It will be N^2/（π^2） times if the number of N is large enough. The numerical simulations confirm the conclusion in quality. This paper also discusses the effects on synchronization by adding one link with different length d to these two different structures. It can be seen that the effects are different. Theoretical results are accordant to numerical simulations. Synchronization is an essential physics problem. These results proposed in this paper have some important reference meanings on the real world networks, such as the bioecological system networks, the designing of the circuit, etc.
基金Supported by the Key Laboratory of Financial Mathematics of Fujian Province University(Putian University)(No.JR202203)NSF of Anhui Province(No.2008085MA06)the project of Anhui Education Committee(No.gxyqZD2019009)。
基金Foundation item： Supported by the Fund of Beijing Education Committee（KM200610005024） Supported by the National Natural Science Foundation of China（10671061）
文摘We consider the sufficient and necessary conditions for the formal triangular matrix ring being right minsymmetric, right DS, semicommutative, respectively.
基金The NSF (10871042,10971024) of Chinathe Specialized Research Fund (200802860024) for the Doctoral Program of Higher Education
文摘A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.
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Supported by the National Natural Science Foundation of China (Grant No.10901129)Lanzhou Jiaotong Daxue Zixuan Keti (Grant No.409039)
文摘Let R be a ring.We show in the paper that the subring Un（R） of the upper triangular matrix ring Tn（R） is α-skew Armendariz if and only if R is α-rigid,also it is maximal in some non α-skew Armendariz rings,where α is a ring endomorphism of R with α（1） = 1.
基金Acknowledgements The authors were grateful to the referees for their constructive comments and suggestions. This work was supported by the National Natural Science Foundation of China （Grant No. 11371109） and the Education Department of Heilongjiang Province of China （No. 12541605）.
文摘Abstract We first consider the group inverses of the block matrices（AB0C）over a weakly finite ring. Then we study the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrices（ABCD）over a ring with unity 1 under the following conditions respectively： （i） B = C, D = 0,B#and（BπA0#both exist; （ii） B is invertible and m = n;（iii）A#and （D - CA#B）# both exist, C = CAA#, where A and D are m × m and n × n matrices, respectively.
文摘An associative ring with identity R is called Armendariz if, whenever （∑^m i=0^aix^i）（∑^n j=0^bjx^j）=0 in R[x],aibj=0 for all i and j. An associative ring with identity is called reduced if it has no non-zero nilpotent elements. In this paper, we define a general reduced ring （with or without identity） and a general Armendariz ring （with or without identity）, and identify a class of maximal general Armendariz subrings of matrix rings over general reduced rings.
文摘Let M be a monoid. Maximal M-Armendariz subrings of upper triangular matrix rings are identified when R is M-Armendariz and reduced. Consequently, new families of M- Armendariz rings are presented.
文摘In this paper we continue the study of various ring theoretic properties of Morita contexts.Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch,completely primary,quasi-duo,2-primal,NI,semiprimitive,projective-free,etc.We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective.
