Let R be a commutative ring with identity,M be an R-module,L(M)denote the set of all submodules of M and g■(M)\{OM}.For any submodule N of M,we set gV^(d)(N)={K∈g:K■N}and gζ^(d)(M)=(GV^(d)(N):N■L(M)).Consider χ...Let R be a commutative ring with identity,M be an R-module,L(M)denote the set of all submodules of M and g■(M)\{OM}.For any submodule N of M,we set gV^(d)(N)={K∈g:K■N}and gζ^(d)(M)=(GV^(d)(N):N■L(M)).Consider χ■L(R)\{R},where L(R)is the set of all ideals of R.We set χV(I)=(J∈x:I■J)and χζ(R)={xV(I):I■L(R)}for any ideal I of R.In this paper,we investigate when,for arbitrary χ and g as above,χζ(R)and gζ^(d)(M)form a topology and a semimodule,respectively.We investigate the structure of gζ^(d)(M)in the case that it is a semimodule.展开更多
Let M be a non-zero module over an associative （not necessarily commutative） ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Spec-s（M） of ...Let M be a non-zero module over an associative （not necessarily commutative） ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Spec-s（M） of second submodules of M and the spectrum Spec-c（M） of coprime submodules of M, study several properties of these spaces and investigate their internlay with the algebraic properties of M.展开更多
In previous work of the author,a top intersection product of toric b-divisors on a smooth complete toric variety is defined.It is shown that a nef toric b-divisor corresponds to a convex set and that its top inetersec...In previous work of the author,a top intersection product of toric b-divisors on a smooth complete toric variety is defined.It is shown that a nef toric b-divisor corresponds to a convex set and that its top inetersection number equals the volume of this convex set.The goal of this article is to extend this result and define an intersection product of sufficiently positive toric b-classes of arbitrary codimension.For this,we extend the polytope algebra of McMullen to the so called convex-set algebra and we show that it embeds in the toric b-Chow group.In this way,the convex-set algebra can be viewed as a ring for an intersection theory for sufficiently positive toric b-classes.As an application,we show that some Hodge type inequalities are satisfied for the convex set algebra.展开更多
We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4.As main ingredients,we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs,an...We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4.As main ingredients,we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs,and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.展开更多
In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey,...In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey, and also a non-commutative generalization of the Zariski spectrum of a commutative ring. But quantum spaces are not good enough to have much of the properties of topological spaces, such as product spaces and quotient spaces.展开更多
In this paper,we discuss some open problems of non-commutative algebra and noncommutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.More exactly,we will analyze the isomorphism ...In this paper,we discuss some open problems of non-commutative algebra and noncommutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.More exactly,we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain.The Serre’s conjecture will be discussed for a particular class of skew PBW extensions.The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings.Advances for the solution of some of the problems are included.展开更多
文摘Let R be a commutative ring with identity,M be an R-module,L(M)denote the set of all submodules of M and g■(M)\{OM}.For any submodule N of M,we set gV^(d)(N)={K∈g:K■N}and gζ^(d)(M)=(GV^(d)(N):N■L(M)).Consider χ■L(R)\{R},where L(R)is the set of all ideals of R.We set χV(I)=(J∈x:I■J)and χζ(R)={xV(I):I■L(R)}for any ideal I of R.In this paper,we investigate when,for arbitrary χ and g as above,χζ(R)and gζ^(d)(M)form a topology and a semimodule,respectively.We investigate the structure of gζ^(d)(M)in the case that it is a semimodule.
文摘Let M be a non-zero module over an associative （not necessarily commutative） ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Spec-s（M） of second submodules of M and the spectrum Spec-c（M） of coprime submodules of M, study several properties of these spaces and investigate their internlay with the algebraic properties of M.
基金Supported by the SFB Higher Invariants at the University of Regensburg。
文摘In previous work of the author,a top intersection product of toric b-divisors on a smooth complete toric variety is defined.It is shown that a nef toric b-divisor corresponds to a convex set and that its top inetersection number equals the volume of this convex set.The goal of this article is to extend this result and define an intersection product of sufficiently positive toric b-classes of arbitrary codimension.For this,we extend the polytope algebra of McMullen to the so called convex-set algebra and we show that it embeds in the toric b-Chow group.In this way,the convex-set algebra can be viewed as a ring for an intersection theory for sufficiently positive toric b-classes.As an application,we show that some Hodge type inequalities are satisfied for the convex set algebra.
基金supported by the China National Postdoctoral Program for Innovative Talents(Grant No.BX2021269)by the China Postdoctoral Science Foundation(Grant No.2021M702925)。
文摘We prove the termination of flips for pseudo-effective NQC log canonical generalized pairs of dimension 4.As main ingredients,we verify the termination of flips for 3-dimensional NQC log canonical generalized pairs,and show that the termination of flips for pseudo-effective NQC log canonical generalized pairs which admit NQC weak Zariski decompositions follows from the termination of flips in lower dimensions.
基金Supported by National Natural Science Foundation of China (Grant No. 10731050)
文摘In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey, and also a non-commutative generalization of the Zariski spectrum of a commutative ring. But quantum spaces are not good enough to have much of the properties of topological spaces, such as product spaces and quotient spaces.
文摘In this paper,we discuss some open problems of non-commutative algebra and noncommutative algebraic geometry from the approach of skew PBW extensions and semi-graded rings.More exactly,we will analyze the isomorphism arising in the investigation of the Gelfand–Kirillov conjecture about the commutation between the center and the total ring of fractions of an Ore domain.The Serre’s conjecture will be discussed for a particular class of skew PBW extensions.The questions about the Noetherianity and the Zariski cancellation property of Artin–Schelter regular algebras will be reformulated for semi-graded rings.Advances for the solution of some of the problems are included.
