Let T be a formal triangular matrix ring.We prove that,if for each 1≤j<i≤n,U_(ij) is flat on both sides,then a left T-module■is Ding projective if and only if M1 is a Ding projective left A1-module and for each ...Let T be a formal triangular matrix ring.We prove that,if for each 1≤j<i≤n,U_(ij) is flat on both sides,then a left T-module■is Ding projective if and only if M1 is a Ding projective left A1-module and for each 1≤k≤n-1 the mappingФk+1,k:U_(k+1),k■AkM_(k)→M_(k)+1 is injective with cokernel Ding projective over Ak+1.As a consequence,we describe Ding projective dimension of a left T-module.展开更多
Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomp...Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomposed into a product of involutive, inner and diagonal automorphisms.展开更多
We consider the sufficient and necessary conditions for the formal triangular matrix ring being right minsymmetric, right DS, semicommutative, respectively.
An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of...An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of R is uniquely strongly clean. The uniquely strong cleanness of the triangular matrix ring is studied. Let R be a local ring. It is shown that any n × n upper triangular matrix ring over R is uniquely strongly clean if and only if R is uniquely bleached and R/J（R） ≈Z2.展开更多
In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely gene...In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely generated projec-tive T-modules to have n in the stable range is given under the assumption that A and B are exchange rings.展开更多
In this paper we study Gorenstein(semiheredity)heredity,finite presentedness and F P-injectivity of modules over a formal triangular matrix ring.We provide necessary and sufficient conditions for such rings to be Gore...In this paper we study Gorenstein(semiheredity)heredity,finite presentedness and F P-injectivity of modules over a formal triangular matrix ring.We provide necessary and sufficient conditions for such rings to be Gorenstein(semihereditary)hereditary and investigate when a triangular matrix ring is an n-FC ring.展开更多
Let R and S be rings with identity, M be a unitary（R, S）-bimodule and T =（R M0 S）be the upper triangular matrix ring determined by R, S and M. In this paper we prove that under certain conditions a Jordan bideriva...Let R and S be rings with identity, M be a unitary（R, S）-bimodule and T =（R M0 S）be the upper triangular matrix ring determined by R, S and M. In this paper we prove that under certain conditions a Jordan biderivation of an upper triangular matrix ring T is a biderivation of T.展开更多
This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problem...This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.展开更多
The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice ...The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.展开更多
Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（...Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（n,F） is said to be a product zero derivation if {φ（x）,y] ＋ [x, φ（y）] = 0 whenever {x, y} = 0,x,y ∈ N（n,F）. In this paper, we prove that a linear map on N（n, F） is a product zero derivation if and only if φ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N（n, F）.展开更多
Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using t...Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.展开更多
Let T=(A0 UB)be a triangular matrix ring with A,B rings and U a B-A-bimodule.We construct resolving subcategories of T-Mod from those of A-Mod and B-Mod.Then we give an estimate of the global resolving resolution dime...Let T=(A0 UB)be a triangular matrix ring with A,B rings and U a B-A-bimodule.We construct resolving subcategories of T-Mod from those of A-Mod and B-Mod.Then we give an estimate of the global resolving resolution dimension of T in terms of that of A and of B.Some applications of these results are given.展开更多
In this paper the sufficient and necessary conditions are given for a formal triangular matrix ring to be right PP, generalized right PP, or semihereditary, respectively.
Let R be an arbitrary commutative ring with identity, and let Nn（R） be the set consisting of all n × n strictly upper triangular matrices over R. In this paper, we give an explicit description of the maps（with...Let R be an arbitrary commutative ring with identity, and let Nn（R） be the set consisting of all n × n strictly upper triangular matrices over R. In this paper, we give an explicit description of the maps（without linearity or additivity assumption） φ ： Nn（R） → Nn（R）satisfying φ（xy） = φ（x）y ＋ xφ（y）. As a consequence, additive derivations and derivations of Nn（R） are also described.展开更多
In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of o...In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of operators, where the operator ?was defined by . In this note, a partial answer for the question is given.展开更多
An operator T on a complex separable infinite dimensional Hilbert space is hypercyclic if there is a vector y∈H such that the orbit Orb(T,y)={y,Ty,T^(2)y,T^(3)y,...}is dense in H.Hypercyclic property and supercyclic ...An operator T on a complex separable infinite dimensional Hilbert space is hypercyclic if there is a vector y∈H such that the orbit Orb(T,y)={y,Ty,T^(2)y,T^(3)y,...}is dense in H.Hypercyclic property and supercyclic proeprty are liable to fail for 2×2 upper triangular operator matrices.In this paper,we aim to explore and characterize the hypercyclicity and the supercyclicity for 2×2 upper triangular operator matrices.We obtain a spectral characterization of the norm-closure of the class of all hypercyclic(supercyclic)operators for 2×2 upper triangular operator matrices.展开更多
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 concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- reg...The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.展开更多
We in this note introduce a new concept,so called strongly J-semiclean ring,that is a generalization of strongly J-clean rings.We first observe the basic properties of strongly J-semiclean rings,constructing typical e...We in this note introduce a new concept,so called strongly J-semiclean ring,that is a generalization of strongly J-clean rings.We first observe the basic properties of strongly J-semiclean rings,constructing typical examples.We next investigate conditions on a local ring R that imply that the upper triangular matrix ring Tn(R)is a strongly J-semiclean ring.Also,the criteria on strong J-semicleanness of 2×2 matrices in terms of a quadratic equation are given.As a consequence,several known results relating to strongly J-clean rings are extended to a more general setting.展开更多
Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generali...Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11861055)。
文摘Let T be a formal triangular matrix ring.We prove that,if for each 1≤j<i≤n,U_(ij) is flat on both sides,then a left T-module■is Ding projective if and only if M1 is a Ding projective left A1-module and for each 1≤k≤n-1 the mappingФk+1,k:U_(k+1),k■AkM_(k)→M_(k)+1 is injective with cokernel Ding projective over Ak+1.As a consequence,we describe Ding projective dimension of a left T-module.
文摘Let T(R) be a two-order upper matrix algebra over the semilocal ring R which is determined by R=F×F where F is a field such that CharF=0. In this paper, we prove that any Jordan automorphism of T(R) can be decomposed into a product of involutive, inner and diagonal automorphisms.
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 National Natural Science Foundation of China(No.10971024)the Specialized Research Fund for the Doctoral Program of Higher Education(No.200802860024)the Natural Science Foundation of Jiangsu Province(No.BK2010393)
文摘An element a of a ring R is called uniquely strongly clean if it is the sum of an idempotent and a unit that commute, and in addition, this expression is unique. R is called uniquely strongly clean if every element of R is uniquely strongly clean. The uniquely strong cleanness of the triangular matrix ring is studied. Let R be a local ring. It is shown that any n × n upper triangular matrix ring over R is uniquely strongly clean if and only if R is uniquely bleached and R/J（R） ≈Z2.
文摘In this paper we study the formal triangular matrix ring T =and give some necessary and sufficient conditions for T to be (strongly) separative, m-fold stable and unit 1-stable. Moreover, a condition for finitely generated projec-tive T-modules to have n in the stable range is given under the assumption that A and B are exchange rings.
基金Supported by the National Natural Science Foundation of China(Grant No.11671126)。
文摘In this paper we study Gorenstein(semiheredity)heredity,finite presentedness and F P-injectivity of modules over a formal triangular matrix ring.We provide necessary and sufficient conditions for such rings to be Gorenstein(semihereditary)hereditary and investigate when a triangular matrix ring is an n-FC ring.
文摘Let R and S be rings with identity, M be a unitary（R, S）-bimodule and T =（R M0 S）be the upper triangular matrix ring determined by R, S and M. In this paper we prove that under certain conditions a Jordan biderivation of an upper triangular matrix ring T is a biderivation of T.
基金supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20070126002)the National Natural Science Foundation of China (No. 10962004)
文摘This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.
基金＊The project supported by the National Key Basic Research Development of China under Grant No. N1998030600 and National Natural Science Foundation of China under Grant No. 10072013
文摘The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.
基金Supported by the National Natural Science Foundation of China(Grant No.11101084)the Natural Science Foundation of Fujian Province(Grant No.2013J01005)
文摘Let F be a field, n ≥ 3, N（n,F） the strictly upper triangular matrix Lie algebra consisting of the n × n strictly upper triangular matrices and with the bracket operation {x, y} = xy-yx. A linear map φ on N（n,F） is said to be a product zero derivation if {φ（x）,y] ＋ [x, φ（y）] = 0 whenever {x, y} = 0,x,y ∈ N（n,F）. In this paper, we prove that a linear map on N（n, F） is a product zero derivation if and only if φ is a sum of an inner derivation, a diagonal derivation, an extremal product zero derivation, a central product zero derivation and a scalar multiplication map on N（n, F）.
文摘Let U be a (B, A)-bimodule, A and B be rings, and be a formal triangular matrix ring. In this paper, we characterize the structure of relative Ding projective modules over T under some conditions. Furthermore, using the left global relative Ding projective dimensions of A and B, we estimate the relative Ding projective dimension of a left T-module.
文摘Let T=(A0 UB)be a triangular matrix ring with A,B rings and U a B-A-bimodule.We construct resolving subcategories of T-Mod from those of A-Mod and B-Mod.Then we give an estimate of the global resolving resolution dimension of T in terms of that of A and of B.Some applications of these results are given.
基金Partially supported by the Fund (KM200610005024) of Beijing Education Committeethe NNSF (10671061) of China.
文摘In this paper the sufficient and necessary conditions are given for a formal triangular matrix ring to be right PP, generalized right PP, or semihereditary, respectively.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1117134311426121)+1 种基金the Science Foundation of Jiangxi University of Science and Technology(Grant Nos.NSFJ2014–K12NSFJ2015–G24)
文摘Let R be an arbitrary commutative ring with identity, and let Nn（R） be the set consisting of all n × n strictly upper triangular matrices over R. In this paper, we give an explicit description of the maps（without linearity or additivity assumption） φ ： Nn（R） → Nn（R）satisfying φ（xy） = φ（x）y ＋ xφ（y）. As a consequence, additive derivations and derivations of Nn（R） are also described.
文摘In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of operators, where the operator ?was defined by . In this note, a partial answer for the question is given.
基金Supported by National Natural Science Foundation of China(Grant No.11671201)。
文摘An operator T on a complex separable infinite dimensional Hilbert space is hypercyclic if there is a vector y∈H such that the orbit Orb(T,y)={y,Ty,T^(2)y,T^(3)y,...}is dense in H.Hypercyclic property and supercyclic proeprty are liable to fail for 2×2 upper triangular operator matrices.In this paper,we aim to explore and characterize the hypercyclicity and the supercyclicity for 2×2 upper triangular operator matrices.We obtain a spectral characterization of the norm-closure of the class of all hypercyclic(supercyclic)operators for 2×2 upper triangular operator matrices.
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The Foundation for Excellent Doctoral Dissertationof Southeast University (NoYBJJ0507)the National Natural ScienceFoundation of China (No10571026)the Natural Science Foundation ofJiangsu Province (NoBK2005207)
文摘The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.
基金Supported by the Excellent Course Project for Postgraduates of Hunan University of Science and Technology(Grant No. J52112)
文摘We in this note introduce a new concept,so called strongly J-semiclean ring,that is a generalization of strongly J-clean rings.We first observe the basic properties of strongly J-semiclean rings,constructing typical examples.We next investigate conditions on a local ring R that imply that the upper triangular matrix ring Tn(R)is a strongly J-semiclean ring.Also,the criteria on strong J-semicleanness of 2×2 matrices in terms of a quadratic equation are given.As a consequence,several known results relating to strongly J-clean rings are extended to a more general setting.
基金National Natural science Foundation of China(10171082)the Cultivation Fund of the Key Scientific Technical Innovation Project,Ministry of Education of ChinaTRAPOYT
文摘Let R be a ring and S a cancellative and torsion-free monoid and 〈 a strict order on S. If either （S,≤） satisfies the condition that 0 ≤ s for all s ∈ S, or R is reduced, then the ring [[R^S,≤]] of the generalized power series with coefficients in R and exponents in S has the same triangulating dimension as R. Furthermore, if R is a PWP ring, then so is [[R^S,≤]].
