Pseudopolar rings are closely related to strongly -regular rings, uniquelystrongly clean rings and semiregular rings. In this paper, we investigate pseudopolar-ity of generalized matrix rings Ks（R） over a local ring...Pseudopolar rings are closely related to strongly -regular rings, uniquelystrongly clean rings and semiregular rings. In this paper, we investigate pseudopolar-ity of generalized matrix rings Ks（R） over a local ring R. We determine the conditionsunder which elements of Ks（R） are pseudopolar. Assume that R is a local ring. It isshown that A ∈ Ks（R） is pseudopolar if and only if A is invertible or A^2 ∈ J（Ks（R））or A is similar to a diagonal matrix [ u 0 0 j ]; where lu -rj and lj-ru are injectiveand u 2 U（R） and j ∈ J（R）. Furthermore, several equivalent conditions for Ks（R）over a local ring R to be pseudopolar are obtained.展开更多
In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probab...In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probabilities of successful impersonation and substitution attack under the hypothesis that the cecoding rules are chosen according to a uniform probability distribution.展开更多
In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R ...In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y∈R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y∈R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his Ph D thesis(Question 6.22 in Sato(2016))and prove that the statement“a commutative ring is local if and only if it has at most one maximal ideal”is equivalent to ACA_0 over RCA_0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA_0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.展开更多
In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ...In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.展开更多
In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We th...In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We then study the deep hole problem of generalized Reed-Solomon (RS) codes over finite local rings. Several different classes of deep holes are constructed. The relationship between finite geometry and deep holes of RS codes over finite local rings are also studied.展开更多
We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal...1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.展开更多
Let R be an associative ring with identity. R is said to be semilocal if R/J（R） is （semisimple） Artinian, where J（R） denotes the Jacobson radical of R. In this paper, we give necessary and sufficient conditions ...Let R be an associative ring with identity. R is said to be semilocal if R/J（R） is （semisimple） Artinian, where J（R） denotes the Jacobson radical of R. In this paper, we give necessary and sufficient conditions for the group ring RG to be semilocal, where G is a locally finite nilpotent group.展开更多
A flattened elliptic ring containing an electron is studied. The emphasis is placed on clarifying the effect of the flattening. The localized states are classified into four types according to their inherent nodes. Wh...A flattened elliptic ring containing an electron is studied. The emphasis is placed on clarifying the effect of the flattening. The localized states are classified into four types according to their inherent nodes. When the ring becomes more flattened, the total probability of dipole absorption of each state is found to be reduced. Furthermore, each spectral line of absorption is found to shift towards red and may split into a few lines, and these lines as a whole become more diffusive.展开更多
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
A ring R is called a GVNL-ring if a or 1-a is π-regular for every a∈R,as a common generalization of local and π-regular rings.It is proved that if R is a GVNL ring,then either(1-e)R(1-e) or eRe is a π-regular ring...A ring R is called a GVNL-ring if a or 1-a is π-regular for every a∈R,as a common generalization of local and π-regular rings.It is proved that if R is a GVNL ring,then either(1-e)R(1-e) or eRe is a π-regular ring for every idempotent e of R.We prove that the center of a GVNL ring is also GVNL and every abelian GVNL ring is SGVNL.The formal power series ring R[x] is GVNL if and only if R is a local ring.展开更多
Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all...Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.展开更多
In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those gra...In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.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
文摘Pseudopolar rings are closely related to strongly -regular rings, uniquelystrongly clean rings and semiregular rings. In this paper, we investigate pseudopolar-ity of generalized matrix rings Ks（R） over a local ring R. We determine the conditionsunder which elements of Ks（R） are pseudopolar. Assume that R is a local ring. It isshown that A ∈ Ks（R） is pseudopolar if and only if A is invertible or A^2 ∈ J（Ks（R））or A is similar to a diagonal matrix [ u 0 0 j ]; where lu -rj and lj-ru are injectiveand u 2 U（R） and j ∈ J（R）. Furthermore, several equivalent conditions for Ks（R）over a local ring R to be pseudopolar are obtained.
基金Foundation item:The Key Project(03060)of Chinese Ministry of Education.
文摘In this paper, we determine the normal forms of idempotent matrices for similarity over finite local rings Z/p^kZ, from which we construct a Cartesian authentication code and compute its size parameters and the probabilities of successful impersonation and substitution attack under the hypothesis that the cecoding rules are chosen according to a uniform probability distribution.
基金supported by National Natural Science Foundation of China(Grant No.12301001)。
文摘In this paper,we study local rings from the perspective of reverse mathematics.We define local rings in a first-order way by usingΠ_2~0 properties of invertible elements,where for a ring R possibly not commutative,R is left(resp.right)local if for any non-left(resp.non-right)invertible elements x,y∈R,x+y is not left(resp.right)invertible;R is local if for any non-invertible elements x,y∈R,x+y is not invertible.Firstly,we solve a question of Sato on characterizations of commutative local rings in his Ph D thesis(Question 6.22 in Sato(2016))and prove that the statement“a commutative ring is local if and only if it has at most one maximal ideal”is equivalent to ACA_0 over RCA_0.We also obtain a nice corollary in computable mathematics,i.e.,there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K.Secondly,we study the equivalence among left local rings,right local rings,and local rings,showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA_0.Finally,we extend the results of reverse mathematics on commutative local rings to noncommutative rings.
基金The first author is supported by Fundamental Research Funds for the Central Universi- ties （No. XDJK2013C060）, Chongqing Research Program of Application Foundation and Advanced Technology （No. cstc2014jcyjA00028） and Scientific Research Foundation for Doctors of Southwest University （No. SWUl12054）. The second author is supported by National Natural Science Foundation of China （No. 11271250）.
文摘In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.
基金The research of Jun Zhang was supported by the National Natural Science Foundation of China(Grant No.11971321)by National Key Research and Development Program of China(Grant No.2018YFA0704703)The research of Haiyan Zhou was supported by the National Natural Science Foundation of China(Grant No.12071221).
文摘In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We then study the deep hole problem of generalized Reed-Solomon (RS) codes over finite local rings. Several different classes of deep holes are constructed. The relationship between finite geometry and deep holes of RS codes over finite local rings are also studied.
基金This research was supported by the National Natural Science Foundation of China(No.11801356,No.11401368,No.11971338)by the Natural Science Foundation of Shanghai(No.19ZR1424100).
文摘We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.
文摘1. Introduction Since O’Meara worked out the automorphisms of orthogonal groups Ωn(V) over fields in [4] by using well-known residual space method, many results about the isomorphisms and automorphisms of orthogonal groups over integral domains have been achieved. Refer to O’Meara, Hahn for example. B. R. McDonald, in [12], determined the automorphisms of O(V) over local rings with 2 a unit by using involutions.
基金Foundation item:The NNSF(10571026)of China,the NSF(BK2005207)of Jiangsu Provincethe Specialized Research Fund(20060286006)for the Doctoral Program of Higher Education.
文摘Let R be an associative ring with identity. R is said to be semilocal if R/J（R） is （semisimple） Artinian, where J（R） denotes the Jacobson radical of R. In this paper, we give necessary and sufficient conditions for the group ring RG to be semilocal, where G is a locally finite nilpotent group.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10574163 and 10675174)
文摘A flattened elliptic ring containing an electron is studied. The emphasis is placed on clarifying the effect of the flattening. The localized states are classified into four types according to their inherent nodes. When the ring becomes more flattened, the total probability of dipole absorption of each state is found to be reduced. Furthermore, each spectral line of absorption is found to shift towards red and may split into a few lines, and these lines as a whole become more diffusive.
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.
基金supported by the grant of National Natural Science Foundation of China(10971024)the Nanjing University of Posts and Telecommunications(NY209022)
文摘A ring R is called a GVNL-ring if a or 1-a is π-regular for every a∈R,as a common generalization of local and π-regular rings.It is proved that if R is a GVNL ring,then either(1-e)R(1-e) or eRe is a π-regular ring for every idempotent e of R.We prove that the center of a GVNL ring is also GVNL and every abelian GVNL ring is SGVNL.The formal power series ring R[x] is GVNL if and only if R is a local ring.
基金Supported by Research Funds of Hubei Province(D20144401,Q20174503)。
文摘Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.
基金Supported by the Natural Sciences Foundation of Guangxi Province(0575052, 0640070)Supported by the Innovation Project of Guangxi Graduate Education(2006106030701M05)Supported by the Scientific Research Foundation of Guangxi Educational Committee(200707LX233
文摘In [1], Joe Warfel investigated the diameter of a zero-divisor graph for a direct product R 1 × R 2 with respect to the diameter of the zero-divisor graph of R 1 and R 2 . But the author only considered those graphs whose diameters ≥ 1 and discussed six cases. This paper further discusses the other nine cases and also gives a complete characterization for the possible diameters for left Artin rings.
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.
