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
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
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 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
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
In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A co...In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.展开更多
Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG(H)={α ∈ RG|αh = hα for all h ∈ H), which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG(H...Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG(H)={α ∈ RG|αh = hα for all h ∈ H), which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG(H) is just the central subalgebra Z(RG) of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG(H). Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG(H) to CRG(H), that is, 7(CRG(H)) = CRG(H).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
Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup&...Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup>⊕K implies m≥n,and R ∈IBN<sub>2</sub> means that R<sup>m</sup>■R<sup>m</sup>⊕K implies K=0.In this paper we give some characteristic properties of IBN<sub>1</sub> and IBN<sub>2</sub>,with orderings o the Grothendieck groups.In addition,we obtain the following results:(1)If R ∈IBM<sub>1</sub> and all finitely generated projective left R-modules are stably free,then the Grothendieck group K<sub>o</sub>(R)is a totally ordered abelian group.(2)If the pre-ordering of the Grothendieck group K<sub>o</sub>(R)of a ring R is a partial ordering,then R ∈IBM<sub>1</sub> or K<sub>o</sub>(R)=0.展开更多
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 K be a finite abelian group and let H be the holomorph of K. It is shown that every Coleman automorphism of H is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalize...Let K be a finite abelian group and let H be the holomorph of K. It is shown that every Coleman automorphism of H is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalizer property holds for H.展开更多
The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z+-modules over the near-group fusion ring K(Z3, 3) are explicitly classified. It turns...The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z+-modules over the near-group fusion ring K(Z3, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible Z+-modules of rank 2 and two inequivalent irreducible Z+-modules of rank 4 over K(Z3, 3).展开更多
In this article we describe the stable behavior of the augmentation quotients Qn (G) for the groups G of order p^5 with even numbers of generators, where p is an odd prime.
Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in wh...Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in which the centralizer of any element is nilpotent; (2) CIT-groups, i.e., groups of even order in which the centralizer of any involution is a 2-group. In particular, the normalizer conjecture holds for both CN-groups and CIT-groups. Additionally, some other results are also obtained.展开更多
Condensation of methyl 7-methylcoumarin-4-acetate(2)with primary amines and with an- thranilic acid gave 7-methyl-2-oxo-N-aryl-2H-[1]-benzopyran-4--acetamide(4a—d)and(7),respectively. Compound 7 underwent cyclization...Condensation of methyl 7-methylcoumarin-4-acetate(2)with primary amines and with an- thranilic acid gave 7-methyl-2-oxo-N-aryl-2H-[1]-benzopyran-4--acetamide(4a—d)and(7),respectively. Compound 7 underwent cyclization to give 2-(7-methyl-2-oxo-2H-[1]-benzopyran-4-yl)-methyl-4H-3,1- benzoxazin-4-one(3).The reaction of 3 with aromatic amines gave the corresponding quinazolone derivatives 5 which tautomerises to the thermodynamically more stable isomer 6,whereas its reaction with Grignard reagents and aromatic aldehydes gave 8a,8b,and 9a,9b,respectively.展开更多
The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called ri...The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called right exp-DR.We investigate the structures of group rings,right quotient rings,matrix rings and(skew)polynomial rings,through the study of right exp-DR rings.In addition,we provide a method of constructing finite non-abelian p-groups for any prime p.展开更多
We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are rig...We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR.We prove that thepolynomial ring over a ring R is right DR if and only if R is commutative.It is also proved that for a prime number p,the group ring KG of a finite p-group G over a field K of characteristic p is right DR if and only if it is right duo,and that there exists a group ring KG that is neither DR nor duo when G is not a p-group.展开更多
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.
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In this paper, we study the basis of augmentation ideals and the quotient groups of finite non-abelian p-group which has a cyclic subgroup of index p, where p is an odd prime, and k is greater than or equal to 3. A concrete basis for the augmentation ideal is obtained and then the structure of its quotient groups can be determined.
基金The NSF(11071155) of Chinathe NSF(2008A03) of Shandong Province
文摘Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG(H)={α ∈ RG|αh = hα for all h ∈ H), which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG(H) is just the central subalgebra Z(RG) of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG(H). Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG(H) to CRG(H), that is, 7(CRG(H)) = CRG(H).
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.
基金Supported by National Nature Science Foundation of China.
文摘Let R be a ring with an identity element.R∈IBN means that R<sup>m</sup>■R<sup>n</sup> implies m=n,R ∈IBN<sub>1</sub> means that R<sup>m</sup> ■R<sup>n</sup>⊕K implies m≥n,and R ∈IBN<sub>2</sub> means that R<sup>m</sup>■R<sup>m</sup>⊕K implies K=0.In this paper we give some characteristic properties of IBN<sub>1</sub> and IBN<sub>2</sub>,with orderings o the Grothendieck groups.In addition,we obtain the following results:(1)If R ∈IBM<sub>1</sub> and all finitely generated projective left R-modules are stably free,then the Grothendieck group K<sub>o</sub>(R)is a totally ordered abelian group.(2)If the pre-ordering of the Grothendieck group K<sub>o</sub>(R)of a ring R is a partial ordering,then R ∈IBM<sub>1</sub> or K<sub>o</sub>(R)=0.
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Supported by National Natural Science Foundation of China(Grant No.11171169)the Doctoral Fund of Shandong Province(Grant No.BS2012SF003)+1 种基金a Project of Shandong Province Higher Educational Science and Technology Program(Grant No.J14LI10)a Project of Shandong Province Higher Educational Excellent Backbone Teachers for International Cooperation and Training
文摘Let K be a finite abelian group and let H be the holomorph of K. It is shown that every Coleman automorphism of H is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalizer property holds for H.
基金This work was supported in part by the National Natural Science Foundation of China (Grant No. 11471282).
文摘The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible Z+-modules over the near-group fusion ring K(Z3, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible Z+-modules of rank 2 and two inequivalent irreducible Z+-modules of rank 4 over K(Z3, 3).
文摘In this article we describe the stable behavior of the augmentation quotients Qn (G) for the groups G of order p^5 with even numbers of generators, where p is an odd prime.
基金Supported by the National Natural Science Foundation of China (71571108), Projects of International (Regional) Cooperation and Exchanges of NSFC (71611530712, 61661136002), Specialized Research Fund for the Doctoral Program of Higher Education of China (20133706110002), Natural Science Foundation of Shandong Province (ZR2015GZ007) Project Funded by China Postdoctoral Science Foundation (2016M590613), Specialized Fund for the Postdoctoral Innovative Research Program of Shandong Province (201602035), Project of Shandong Province Higher Educational Science and Technology Program (J14LI10) and Project of Shandong Province Higher Edu- cational Excellent Backbone Teachers for International Cooperation and Training.
文摘Let G be a finite group. It is proved that any class-preserving Coleman automorphism of G is an inner automorphism whenever G belongs to one of the following two classes of groups: (1) CN-groups, i.e., groups in which the centralizer of any element is nilpotent; (2) CIT-groups, i.e., groups of even order in which the centralizer of any involution is a 2-group. In particular, the normalizer conjecture holds for both CN-groups and CIT-groups. Additionally, some other results are also obtained.
文摘Condensation of methyl 7-methylcoumarin-4-acetate(2)with primary amines and with an- thranilic acid gave 7-methyl-2-oxo-N-aryl-2H-[1]-benzopyran-4--acetamide(4a—d)and(7),respectively. Compound 7 underwent cyclization to give 2-(7-methyl-2-oxo-2H-[1]-benzopyran-4-yl)-methyl-4H-3,1- benzoxazin-4-one(3).The reaction of 3 with aromatic amines gave the corresponding quinazolone derivatives 5 which tautomerises to the thermodynamically more stable isomer 6,whereas its reaction with Grignard reagents and aromatic aldehydes gave 8a,8b,and 9a,9b,respectively.
文摘The object of this article is to initiate the study of a class of rings in which the right duo property is applied in relation to powers of elements and the monoid of all regular elements.Such rings shall be called right exp-DR.We investigate the structures of group rings,right quotient rings,matrix rings and(skew)polynomial rings,through the study of right exp-DR rings.In addition,we provide a method of constructing finite non-abelian p-groups for any prime p.
文摘We study the right duo property on regular elements,and we say that rings with this property are right DR.It is first shown that the right duo property is preserved by right quotient rings when the given rings are right DR.We prove that thepolynomial ring over a ring R is right DR if and only if R is commutative.It is also proved that for a prime number p,the group ring KG of a finite p-group G over a field K of characteristic p is right DR if and only if it is right duo,and that there exists a group ring KG that is neither DR nor duo when G is not a p-group.