The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we inv...The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity and the diameter of Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.展开更多
A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute...A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.展开更多
A ring R with unity is called semiclean, if each of its elements is the sum of a unit and a periodic. Every clean ring is semiclean. It is not easy to characterize a semiclean group ring in general. Our purpose is to ...A ring R with unity is called semiclean, if each of its elements is the sum of a unit and a periodic. Every clean ring is semiclean. It is not easy to characterize a semiclean group ring in general. Our purpose is to consider the following question： If G is a locally finite group or a cyclic group of order 3, then when is RG semiclean？ Some known results on clean group rings are generalized.展开更多
In this article,we present the multiplicative Jordan decomposition in integral group ring of group K8 × C5,where K8 is the quaternion group of order 8.Thus,we give a positive answer to the question raised by Hale...In this article,we present the multiplicative Jordan decomposition in integral group ring of group K8 × C5,where K8 is the quaternion group of order 8.Thus,we give a positive answer to the question raised by Hales A W,Passi I B S and Wilson L E in the paper 'The multiplicative Jordan decomposition in group rings II.展开更多
In this paper, we determine the Jacobson radicals and Brown-McCoy radicals of group rings of certain non-abelian groups and generalize some known results.
Let R ＊θ G be the skew group ring with a F.C group G and the group homomrphism 8 from G to Aut（R）, the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R ＊θ G ...Let R ＊θ G be the skew group ring with a F.C group G and the group homomrphism 8 from G to Aut（R）, the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R ＊θ G will be Noetherian is given, which generalizes the results of LG. connel.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
In this paper, Let R is a ring, G be a finite group of ring automorphisms of R. R*G denote the skew group ring of R under G. We investigate the right p.q.-Baer property of skew group rings under finite group action, A...In this paper, Let R is a ring, G be a finite group of ring automorphisms of R. R*G denote the skew group ring of R under G. We investigate the right p.q.-Baer property of skew group rings under finite group action, Assume that R is a semiprime ring with a finite group G of X-outer ring automorphisms of R, then 1) R*G is p.q.-Baer if and only if R is G-p.q.-Baer;2) if R is p.q.-Baer, then R*G is p.q.-Baer.展开更多
The Tarski theorems, proved by Myasnikov and Kharlampovich and inde-pendently by Sela say that all nonabelian free groups satisfy the same first-order or elementary theory. Kharlampovich and Myasnikov also prove that ...The Tarski theorems, proved by Myasnikov and Kharlampovich and inde-pendently by Sela say that all nonabelian free groups satisfy the same first-order or elementary theory. Kharlampovich and Myasnikov also prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. In this note we examine and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers Z then R[G] is universally equivalent to Z[F] again with respect to an ap-propriate language.展开更多
In this paper, the (quasi-)Baerness of skew group ring and fixed ring is investigated. The following two results are obtained: if R is a simple ring with identity and G an outer automorphism group, then R G is a Baer ...In this paper, the (quasi-)Baerness of skew group ring and fixed ring is investigated. The following two results are obtained: if R is a simple ring with identity and G an outer automorphism group, then R G is a Baer ring;if R is an Artinian simple ring with identity and G an outer automorphism group, then RG is a Baer ring. Moreover, by decomposing Morita Context ring and Morita Context Theory, we provided several conditions of Morita Context ring, which is formed of skew group ring and fixed ring, to be (quasi-)Baer ring.展开更多
Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under...Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under the assumption that R^σ G is a,separable extension over R^σ H,it is shown that R^σ G and R^σ H share the same homological dimensions.Several known results are then obtained as corollaries.Moreover,we investigate the relationships between the homological dimensions of Ra G and the homological dimensions of a commutative ring R,using the trivial R^σ G-module.展开更多
Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given.Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised.
Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessa...Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessary and sufficient conditions are given on the group G, its involution, the ring R and the extension for the set of skew-symmetric elements to be commutative and for it to be anticommutative.展开更多
A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that...A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.展开更多
The title compound, [Ni（tssb）（2,2-bipy）2].5（H2O） 1 （tssbH2 =2-[（E）-（2-oxido- phenyl）methyleneamino]ethanesulfonato, 2,2-bipy = 2,2＇-bipyridinyl）, belongs to orthorhombic, space group Pbcn with a = 20.3983...The title compound, [Ni（tssb）（2,2-bipy）2].5（H2O） 1 （tssbH2 =2-[（E）-（2-oxido- phenyl）methyleneamino]ethanesulfonato, 2,2-bipy = 2,2＇-bipyridinyl）, belongs to orthorhombic, space group Pbcn with a = 20.3983（18）, b = 17.6929（15）, c = 17.0897（15） nm, V= 6167.8（9） nm^3, Mr= 688.38, Z = 8, De = 1.481 g.cm^-3, F（000） = 2880,μ = 0.758 mm-1 and S =1.099. Each NiIr atom is six-coordinated by one N and one O atoms from one tssb^2- anion and four N atoms from two 2,2-bipy ligands to give a distorted octahedral geometry. Noticeably, there exists a rare octa-mem- bered water ring which presents a 1D chain by sulfonic group.展开更多
It is well known that K<sub>0</sub>R（?）Z（?）（?）<sub>0</sub>R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）<sub>0</sub>R...It is well known that K<sub>0</sub>R（?）Z（?）（?）<sub>0</sub>R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）<sub>0</sub>R. On the other hand, linear representations of groups can be seen as the finitely generated projective modules over group rings. Thus, it is very useful to study the properties of reduced groups of group rings.展开更多
Quantum algorithms bring great challenges to classical public key cryptosystems, which makes cryptosystems based on non-commutative algebraic systems hop topic. The braid groups, which are non-commutative, have attrac...Quantum algorithms bring great challenges to classical public key cryptosystems, which makes cryptosystems based on non-commutative algebraic systems hop topic. The braid groups, which are non-commutative, have attracted much attention as a new platform for constructing quantum attack-resistant cryptosystems. A ring signature scheme is proposed based on the difficulty of the root extraction problem over braid groups, which can resist existential forgery against the adaptively cho-sen-message attack under the random oracle model.展开更多
A ring R is called right principally quasi-Baer (simply, right p.q.-Baer) if the right annihilator of every principal right ideal of R is generated by an idempotent. For a ring R, let G be a finite group of ring autom...A ring R is called right principally quasi-Baer (simply, right p.q.-Baer) if the right annihilator of every principal right ideal of R is generated by an idempotent. For a ring R, let G be a finite group of ring automorphisms of R. We denote the fixed ring of R under G by RG. In this work, we investigated the right p.q.-Baer property of fixed rings under finite group action. Assume that R is a semiprime ring with a finite group G of X-outer ring automorphisms of R. Then we show that: 1) If R is G-p.q.-Baer, then RG is p.q.-Baer;2) If R is p.q.-Baer, then RG are p.q.-Baer.展开更多
The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K...The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.展开更多
Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to cons...Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to construct public-key cryptographic protocols. In this article, we propose a new authenticated group key agreement protocol which works in non-abelian near-rings. We have proved that our protocol meets the security attributes under the assumption that the twist conjugacy search problem（TCSP） is hard in near-ring.展开更多
基金The NSF(10971024)of Chinathe Specialized Research Fund(200802860024)for the Doctoral Program of Higher Educationthe NSF(BK2010393)of Jiangsu Province
文摘The commuting graph of an arbitrary ring R, denoted by Г（R）, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity and the diameter of Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.
文摘A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely strongly clean （USC for short） if every element is uniquely the sum of an idempotent and a unit that commute. In this article, some conditions on a ring R and a group G such that RG is clean are given. It is also shown that if G is a locally finite group, then the group ring RG is USC if and only if R is USC, and G is a 2-group. The left uniquely exchange group ring, as a middle ring of the uniquely clean ring and the USC ring, does not possess this property, and so does the uniquely exchange group ring.
基金Supported by the National Natural Science Foundation of China(Grant No.11401009)the Natural Science Foundation of Ahui Province(Grant No.1408085QA01)
文摘A ring R with unity is called semiclean, if each of its elements is the sum of a unit and a periodic. Every clean ring is semiclean. It is not easy to characterize a semiclean group ring in general. Our purpose is to consider the following question： If G is a locally finite group or a cyclic group of order 3, then when is RG semiclean？ Some known results on clean group rings are generalized.
文摘In this article,we present the multiplicative Jordan decomposition in integral group ring of group K8 × C5,where K8 is the quaternion group of order 8.Thus,we give a positive answer to the question raised by Hales A W,Passi I B S and Wilson L E in the paper 'The multiplicative Jordan decomposition in group rings II.
文摘In this paper, we determine the Jacobson radicals and Brown-McCoy radicals of group rings of certain non-abelian groups and generalize some known results.
基金Supported by the NSF of Educational Department of Henan Province(20025100003)
文摘Let R ＊θ G be the skew group ring with a F.C group G and the group homomrphism 8 from G to Aut（R）, the group of automorphisms of the ring R. In this paper,the necessary and sufficient condition such that R ＊θ G will be Noetherian is given, which generalizes the results of LG. connel.
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.
文摘In this paper, Let R is a ring, G be a finite group of ring automorphisms of R. R*G denote the skew group ring of R under G. We investigate the right p.q.-Baer property of skew group rings under finite group action, Assume that R is a semiprime ring with a finite group G of X-outer ring automorphisms of R, then 1) R*G is p.q.-Baer if and only if R is G-p.q.-Baer;2) if R is p.q.-Baer, then R*G is p.q.-Baer.
文摘The Tarski theorems, proved by Myasnikov and Kharlampovich and inde-pendently by Sela say that all nonabelian free groups satisfy the same first-order or elementary theory. Kharlampovich and Myasnikov also prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. In this note we examine and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers Z then R[G] is universally equivalent to Z[F] again with respect to an ap-propriate language.
文摘In this paper, the (quasi-)Baerness of skew group ring and fixed ring is investigated. The following two results are obtained: if R is a simple ring with identity and G an outer automorphism group, then R G is a Baer ring;if R is an Artinian simple ring with identity and G an outer automorphism group, then RG is a Baer ring. Moreover, by decomposing Morita Context ring and Morita Context Theory, we provided several conditions of Morita Context ring, which is formed of skew group ring and fixed ring, to be (quasi-)Baer ring.
基金supported by the Scientific Research Foundation of Hunan Provincial Education Department(no.18C0997).
文摘Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under the assumption that R^σ G is a,separable extension over R^σ H,it is shown that R^σ G and R^σ H share the same homological dimensions.Several known results are then obtained as corollaries.Moreover,we investigate the relationships between the homological dimensions of Ra G and the homological dimensions of a commutative ring R,using the trivial R^σ G-module.
基金Supported by China Natural Science Funds(60075016)Guangxi Natural Science Funds(0135005)Guangxi Selected Experts Funds
文摘Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given.Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised.
文摘Given an involution in a group G, it can be extended in various ways to an involution in the group ring RG, where R is a ring, not necessarily commutative. In this paper nonlinear extensions are considered and necessary and sufficient conditions are given on the group G, its involution, the ring R and the extension for the set of skew-symmetric elements to be commutative and for it to be anticommutative.
基金This research was supported in part by the National Natural Science Foundation of China （11371089, 11201064）, the Specialized Research Fund for the Doctoral Program of Higher Education （20120092110020）, the Natural Science Foundation of Jiangsu Province （BK20130599）, and a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
文摘A ring with involution ＊ is called ＊-clean if each of its elements is the sum of a unit and a projection. It is obvious that ＊-clean rings are clean. Vas asked whether there exists a clean ring with involution that is not ＊-clean. In this paper, we investigate when a group ring RG is ＊-clean, where ＊ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ＊-clean, where R is a commutative local ring and G is one of the groups C3,C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ＊-clean.
基金Supported by the Key Laboratory of Non-ferrous Metal Materials and New Processing TechnologyMinistry of Education and the State Key Laboratory of Coordination Chemistry
文摘The title compound, [Ni（tssb）（2,2-bipy）2].5（H2O） 1 （tssbH2 =2-[（E）-（2-oxido- phenyl）methyleneamino]ethanesulfonato, 2,2-bipy = 2,2＇-bipyridinyl）, belongs to orthorhombic, space group Pbcn with a = 20.3983（18）, b = 17.6929（15）, c = 17.0897（15） nm, V= 6167.8（9） nm^3, Mr= 688.38, Z = 8, De = 1.481 g.cm^-3, F（000） = 2880,μ = 0.758 mm-1 and S =1.099. Each NiIr atom is six-coordinated by one N and one O atoms from one tssb^2- anion and four N atoms from two 2,2-bipy ligands to give a distorted octahedral geometry. Noticeably, there exists a rare octa-mem- bered water ring which presents a 1D chain by sulfonic group.
文摘It is well known that K<sub>0</sub>R（?）Z（?）（?）<sub>0</sub>R, where R is a commutative ring. So the Grothendieck group of R can be given by the reduced group （?）<sub>0</sub>R. On the other hand, linear representations of groups can be seen as the finitely generated projective modules over group rings. Thus, it is very useful to study the properties of reduced groups of group rings.
基金Supported by the National Natural Science Foundation of China (No. 10501053)
文摘Quantum algorithms bring great challenges to classical public key cryptosystems, which makes cryptosystems based on non-commutative algebraic systems hop topic. The braid groups, which are non-commutative, have attracted much attention as a new platform for constructing quantum attack-resistant cryptosystems. A ring signature scheme is proposed based on the difficulty of the root extraction problem over braid groups, which can resist existential forgery against the adaptively cho-sen-message attack under the random oracle model.
文摘A ring R is called right principally quasi-Baer (simply, right p.q.-Baer) if the right annihilator of every principal right ideal of R is generated by an idempotent. For a ring R, let G be a finite group of ring automorphisms of R. We denote the fixed ring of R under G by RG. In this work, we investigated the right p.q.-Baer property of fixed rings under finite group action. Assume that R is a semiprime ring with a finite group G of X-outer ring automorphisms of R. Then we show that: 1) If R is G-p.q.-Baer, then RG is p.q.-Baer;2) If R is p.q.-Baer, then RG are p.q.-Baer.
文摘The present note determines the structure of the K2-group and of its subgroup over a finite commutative ring R by considering relations between R andfinite commutative local ring Ri (1 < i < m), where R Ri and K2(R) =K2(Ri). We show that if charKi= p (Ki denotes the residual field of Ri), then K2(Ri) and its subgroups must be p-groups.
文摘Nowadays some promising authenticated group key agreement protocols are constructed on braid groups, dynamic groups, pairings and bilinear pairings. Hence the non-abelian structure has attracted cryptographers to construct public-key cryptographic protocols. In this article, we propose a new authenticated group key agreement protocol which works in non-abelian near-rings. We have proved that our protocol meets the security attributes under the assumption that the twist conjugacy search problem（TCSP） is hard in near-ring.
