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
We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R...We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.展开更多
In this article,we extend the Zassenhaus lemma(also called the butterfly lemma)and the Schreier refinement theorem to the case of gyrogroups.In addition,we use these results to prove the Jordan–Hölder theorem fo...In this article,we extend the Zassenhaus lemma(also called the butterfly lemma)and the Schreier refinement theorem to the case of gyrogroups.In addition,we use these results to prove the Jordan–Hölder theorem for gyrogroups as well as some theorems regarding subgyrogroup lattices.展开更多
We apply the method of Grobner-Shirshov bases for replicated algebras devel-oped by Kolesnikov to offer a general approach for constructing free products of associative trialgebras(or trioids).In particular,the open p...We apply the method of Grobner-Shirshov bases for replicated algebras devel-oped by Kolesnikov to offer a general approach for constructing free products of associative trialgebras(or trioids).In particular,the open problem of Zhuchok on constructing free products of trioids is solved.展开更多
Let Z_(4) be the ring of integers modulo 4.We study the Λ-constacyclic and(θ,Λ)-cyclic codes over the non-chain ring R=Z_(4)[u,v]/<u^(2)=1,v^(2)=0,uv=vu=0>for a unit Λ=1+2u+2v in R.We define several Gray map...Let Z_(4) be the ring of integers modulo 4.We study the Λ-constacyclic and(θ,Λ)-cyclic codes over the non-chain ring R=Z_(4)[u,v]/<u^(2)=1,v^(2)=0,uv=vu=0>for a unit Λ=1+2u+2v in R.We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over Z_(4) are cyclic,quasi-cyclic or permutation equivalent to this code.For an odd positive integer n,we determine the generator polynomials of cyclic and Λ-constacyclic codes of length n over R.Further,we prove that a(θ,Λ)-cyclic code of length n is a Λ-constacyclic code if n is odd,and a Λ-quasi-twisted code if n is even.A few examples are also incorporated,in which two parameters are new and one is best known to date.展开更多
We classify modules which are free of rank 1 when restricted to the negative part for the Euclidean Lie algebra.Sufficient and necessary conditions for the simplicity of these modules are given.From these modules we c...We classify modules which are free of rank 1 when restricted to the negative part for the Euclidean Lie algebra.Sufficient and necessary conditions for the simplicity of these modules are given.From these modules we construct modules which are infinitely generated when restricted to the maximal toral subalgebra.展开更多
The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this pa...The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.展开更多
The coordinate ring O_(q)(K^(n))of quantum affine space is the K-algebra presented by generators x_(1),...,x_(n) and relations x_(i)x_(j)=q_(ij)a_(j)a_(i) for all i,j.We construct simple O_(q)(K^(n))-modules in a more...The coordinate ring O_(q)(K^(n))of quantum affine space is the K-algebra presented by generators x_(1),...,x_(n) and relations x_(i)x_(j)=q_(ij)a_(j)a_(i) for all i,j.We construct simple O_(q)(K^(n))-modules in a more general setting where the parameters qij lie in a torsion subgroup of K^(*)and show that analogous results hold as in the uniparameter case.展开更多
The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In this pap...The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In this paper,we study certain results on the metric dimension,upper dimension and resolving number of extended annihilating-ideal graph EAG(R)associated to a commutative ring R,denoted by dim M(EAG(R)),dim+(EAG(R))and res(EAG(R)),respectively.Here we prove the finiteness conditions of dim M(EAG(R))and dim+(EAG(R)).In addition,we characterize dim M(EAG(R)),dim+(EAG(R))and res(EAG(R))for artinian rings and the direct product of rings.展开更多
Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions...Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions.Finally,we give some examples to illustrate that both the upper and lower bounds can be reached.展开更多
We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices.Many known criteria for non-congruent numbers can be viewed as special cases of our results.
A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(...A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.展开更多
Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in parti...Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in particular,R is a symmetric ring if and only if R is a(1,1)-symmetric ring.We show that e∈E(R)is left semicentral if and only if R is a(1−e,e)-symmetric ring;in particular,R is an Abel ring if and only if R is a(1−e,e)-symmetric ring for each e∈E(R).We also show that R is(g,e)-symmetric if and only if ge∈E(R),geRge is symmetric,and gxye=gxeye=gxgye for any x,y∈R.Using e-symmetric rings,we construct some new classes of rings.展开更多
Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how ...Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how to construct left cotorsion pais from left n-cotorsion pairs,and establish a one-to-one correspondence between left Frobenius pairs and left(weak)Auslander-Buchweitz contexts.Some applications are given in the Gorenstein homological theory of triangulated categories.展开更多
We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl grou...We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl groups of restricted Cartan typeLiealgebra.展开更多
Aims and Scope.Algebra Colloquium is an international mathematical journal founded at the beginning of 1994.It is edited by Academy of Mathematics&Systems Science,Chinese Academy of Sciences,jointly with Suzhou Un...Aims and Scope.Algebra Colloquium is an international mathematical journal founded at the beginning of 1994.It is edited by Academy of Mathematics&Systems Science,Chinese Academy of Sciences,jointly with Suzhou University,and published quarterly in English in every March,June,September and December.Algebra Colloquium carries original research articles of high level in the field of pure and applied algebra.展开更多
We study the convolution algebra H_(*)^(G×C^(*))(Z)of G×C^(*)-equivariant homology group on the Steinberg variety of type B/C and define an algebra Y that maps to H_(*)^(G×C^(*))(Z).We also study the G-...We study the convolution algebra H_(*)^(G×C^(*))(Z)of G×C^(*)-equivariant homology group on the Steinberg variety of type B/C and define an algebra Y that maps to H_(*)^(G×C^(*))(Z).We also study the G-equivariant case and prove that there is a map from a certain twisted current algebra to H_(*)^(G)(Z).展开更多
Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we giv...Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.展开更多
In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_...In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_(5).展开更多
The natural filtrations of infinite-dimensional Lie superalgebras of O-type over a field of positive characteristic are proved to be invariant under automorphisms by studying ad-nilpotent elements.We observe that two ...The natural filtrations of infinite-dimensional Lie superalgebras of O-type over a field of positive characteristic are proved to be invariant under automorphisms by studying ad-nilpotent elements.We observe that two Lie superalgebras of O-type are isomorphic if and only if the integers in the definition of these Lie superalgebras are intrinsic.Thereby,all the Lie superalgebras of O-type are classified in the sense of isomorphism.展开更多
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.
文摘We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.
文摘In this article,we extend the Zassenhaus lemma(also called the butterfly lemma)and the Schreier refinement theorem to the case of gyrogroups.In addition,we use these results to prove the Jordan–Hölder theorem for gyrogroups as well as some theorems regarding subgyrogroup lattices.
基金supported by the Natural Science Foundation of Huizhou University(2022JB035,HZU202001)the Guangdong Basic and Applied Basic Research Foundation(2023A1515011690)+3 种基金the Characteristic Innovation Project of Guangdong Provincial Department of Education(2023KTSCX145)supported by the NNSF of China(11571121,12071156)supported by the NNSF of China(12101248)by the China Postdoctoral Science Foundation(2021M691099).
文摘We apply the method of Grobner-Shirshov bases for replicated algebras devel-oped by Kolesnikov to offer a general approach for constructing free products of associative trialgebras(or trioids).In particular,the open problem of Zhuchok on constructing free products of trioids is solved.
文摘Let Z_(4) be the ring of integers modulo 4.We study the Λ-constacyclic and(θ,Λ)-cyclic codes over the non-chain ring R=Z_(4)[u,v]/<u^(2)=1,v^(2)=0,uv=vu=0>for a unit Λ=1+2u+2v in R.We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over Z_(4) are cyclic,quasi-cyclic or permutation equivalent to this code.For an odd positive integer n,we determine the generator polynomials of cyclic and Λ-constacyclic codes of length n over R.Further,we prove that a(θ,Λ)-cyclic code of length n is a Λ-constacyclic code if n is odd,and a Λ-quasi-twisted code if n is even.A few examples are also incorporated,in which two parameters are new and one is best known to date.
基金supported by NSF of China(11801117,11971315)the Natural Science Foundation of Guangdong Province,China(2018A030313268)partially supported by NSFof China(11801390).
文摘We classify modules which are free of rank 1 when restricted to the negative part for the Euclidean Lie algebra.Sufficient and necessary conditions for the simplicity of these modules are given.From these modules we construct modules which are infinitely generated when restricted to the maximal toral subalgebra.
基金the support of MATRICS Grant(MTR/2018/000779)funded by SERB,India.
文摘The enhanced power graph P_(E)(S)of a semigroup S is a simple graph whose vertex set is S and two vertices a,y∈S are adjacent if and only if c,y∈(z)for some z∈S,where(z)is the subsemigroup generated by z.In this paper,we first describe the structure of P_(E)(S)for an arbitrary semigroup S,and then discuss the connectedness of P_(E)(S).Further,we characterize the semigroup S in the cases when P_(E)(S)is separately a complete,bipartite,regular,tree and null graph.The planarity,together with the minimum degree and independence number,of P_(E)(S)is also investigated.The chromatic number of a spanning subgraph,i.e.,the cyclic graph,of P_(E)(S)is proved to be countable.In the final part of this paper,we construct an example of a semigroup S such that the chromatic number of P_(E)(S)need not be countable.
文摘The coordinate ring O_(q)(K^(n))of quantum affine space is the K-algebra presented by generators x_(1),...,x_(n) and relations x_(i)x_(j)=q_(ij)a_(j)a_(i) for all i,j.We construct simple O_(q)(K^(n))-modules in a more general setting where the parameters qij lie in a torsion subgroup of K^(*)and show that analogous results hold as in the uniparameter case.
文摘The metric dimension problem is called navigation problem due to its application to robot navigation in space.Further this concept has wide applications in motion planning,sonar and loran station,and so on.In this paper,we study certain results on the metric dimension,upper dimension and resolving number of extended annihilating-ideal graph EAG(R)associated to a commutative ring R,denoted by dim M(EAG(R)),dim+(EAG(R))and res(EAG(R)),respectively.Here we prove the finiteness conditions of dim M(EAG(R))and dim+(EAG(R)).In addition,we characterize dim M(EAG(R)),dim+(EAG(R))and res(EAG(R))for artinian rings and the direct product of rings.
基金supported by the National Natural Science Foundation of China(11971255,11901567,12071120)supported by the Hunan Provincial Natural Science Foundation of China(2023JJ30008)the National Natural Science Foundation of China(12371034).
文摘Let A be a finite dimensional k-algebra and T be a supportτ-tilting right A-module.In this note,we give lower and upper bounds for the global dimension of the endomorphism algebra End_(A)(T)under some mild conditions.Finally,we give some examples to illustrate that both the upper and lower bounds can be reached.
基金supported by NSFC(Nos.12231009,11971224,12071209).
文摘We give some sufficient conditions for non-congruent numbers in terms of the Monsky matrices.Many known criteria for non-congruent numbers can be viewed as special cases of our results.
基金supported by ZJNSF(LY19A010011)NSFC(11971141,12371017)supported by NSFC(11971449,12131015,12371042).
文摘A Clifford deformation of a Koszul Frobenius algebra E is a finite dimensional Z_(2)-graded algebra E(θ),which corresponds to a noncommutative quadric hypersurface E^(!)/(z)for some central regular element z∈E_(2)^(!).It turns out that the bounded derived category D^(b)(gr_(Z_(2))E(θ))is equivalent to the stable category of the maximal Cohen-Macaulay modules over E^(!)/(z)provided that E!is noetherian.As a consequence,E^(!)/(z)is a noncommutative isolated singularity if and only if the corresponding Clifford deformation E(θ)is a semisimple Z_(2)-graded algebra.The preceding equivalence of triangulated categories also indicates that Clifford deformations of trivial extensions of a Koszul Frobenius algebra are related to Knörrer's periodicity theorem for quadric hypersurfaces.As an application,we recover Knörrer's periodicity theorem without using matrix factorizations.
基金supported by the Foundation of Natural Science of China(12301029,11171291)Natural Science Fund for Colleges and Universities in Jiangsu Province(11KJB110019 and 15KJB110023).
文摘Let R be a ring and e,g in E(R),the set of idempotents of R.Then R is called(g,e)-symmetric if abc=0 implies gacbe=0 for any a,b,c∈R.Clearly,R is an e-symmetric ring if and only if R is a(1,e)-symmetric ring;in particular,R is a symmetric ring if and only if R is a(1,1)-symmetric ring.We show that e∈E(R)is left semicentral if and only if R is a(1−e,e)-symmetric ring;in particular,R is an Abel ring if and only if R is a(1−e,e)-symmetric ring for each e∈E(R).We also show that R is(g,e)-symmetric if and only if ge∈E(R),geRge is symmetric,and gxye=gxeye=gxgye for any x,y∈R.Using e-symmetric rings,we construct some new classes of rings.
基金supported by the NSF of China(12001168,11901341,11971225,12171207)Henan University of Engineering(DKJ2019010)+1 种基金the Key Research Project of Education Department of Henan Province(21A110006)Youth Innovation Team of Universities of Shandong Province(2022KJ314).
文摘Let T be a triangulated category with a proper classξof triangles.We introduce the notions of left Frobenius pairs,left(n-)cotorsion pairs and left(weak)Auslander-Buchweitz contexts with respect toξin T.We show how to construct left cotorsion pais from left n-cotorsion pairs,and establish a one-to-one correspondence between left Frobenius pairs and left(weak)Auslander-Buchweitz contexts.Some applications are given in the Gorenstein homological theory of triangulated categories.
基金supported by NSFC(No.12101544)Fundamental Research Funds of Yunnan Province(No.202301AT070415).
文摘We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl groups of restricted Cartan typeLiealgebra.
文摘Aims and Scope.Algebra Colloquium is an international mathematical journal founded at the beginning of 1994.It is edited by Academy of Mathematics&Systems Science,Chinese Academy of Sciences,jointly with Suzhou University,and published quarterly in English in every March,June,September and December.Algebra Colloquium carries original research articles of high level in the field of pure and applied algebra.
基金partially supported by the Fundamental Research Funds for the central universities.
文摘We study the convolution algebra H_(*)^(G×C^(*))(Z)of G×C^(*)-equivariant homology group on the Steinberg variety of type B/C and define an algebra Y that maps to H_(*)^(G×C^(*))(Z).We also study the G-equivariant case and prove that there is a map from a certain twisted current algebra to H_(*)^(G)(Z).
基金supported by the National Natural Science Foundation of China(Grant No.12371041).
文摘Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed feld of characteristic zero.We describe all annihilator ideals of indecomposable H-modules by generators.In particular,we give the classification of all ideals of the finite-dimensional pointed rank one Hopf algebra of nilpotent type over the Klein 4-group.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.2232021G-13).
文摘In this paper,we prove that the electrical Lie algebra e D_(5)is isomorphic to the semidirect product of sp_(4)and a 2-step nilpotent Lie algebra.Furthermore,we classify the irreducible highest weight modules for e D_(5).
基金Supported by the National Natural Science Foundation of China(11501274)the General Project for Educational Department of Liaoning Province,China(LJKMZ20220454,LJKZ0096).
文摘The natural filtrations of infinite-dimensional Lie superalgebras of O-type over a field of positive characteristic are proved to be invariant under automorphisms by studying ad-nilpotent elements.We observe that two Lie superalgebras of O-type are isomorphic if and only if the integers in the definition of these Lie superalgebras are intrinsic.Thereby,all the Lie superalgebras of O-type are classified in the sense of isomorphism.