Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules ...Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.展开更多
Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respec...Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetric α-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α（1） = 1, then R is a symmetric α-ring if and only if R[x]/（x^n） is a symmetric q^--ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q（R） the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q（R） is a symmetric α-ring. Among others we also show that if a ring R is weakly 2-primal and （α, δ）-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric α^--ring.展开更多
We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left ...We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.展开更多
In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and con...In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and construct typical examples.We next describe all Σ-associated primes of the Ore extension R[x; α,δ],the skew Laurent polynomial ring R[x,x-1; α] and the skew power series ring R[[x; α]],in terms of the Σ-associated primes of R in a very straightforward way.Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.展开更多
We focus our attention to the set Gr（ξ） of grouplike elements of a coring ξover a ring A. We do some observations on the actions of the groups U（A） and Aut（ξ~） of units of A and of automorphisms of corings of...We focus our attention to the set Gr（ξ） of grouplike elements of a coring ξover a ring A. We do some observations on the actions of the groups U（A） and Aut（ξ~） of units of A and of automorphisms of corings of ξ, respectively, on Gr（ξ）, and on the subset Gal（ξ）of all Galois grouplike elements. Among them, we give conditions on ξ under which Gal（ξ） is a group, in such a way that there is an exact sequence of groups {1} → U（Ag） → U（A） → Gal（ξ） → {1}, where Ag is the subalgebra of coinvaxiants for some g ∈ Gal（ξ）.展开更多
This paper gives some results on Strong-Armendariz rings and the Ore-extensions R[x,x^-1;α] of Bare, PP and PS rings. And the main two results are： （1） R is a Bear （PP） ring if and only if R[[x]] is a Baer （PP...This paper gives some results on Strong-Armendariz rings and the Ore-extensions R[x,x^-1;α] of Bare, PP and PS rings. And the main two results are： （1） R is a Bear （PP） ring if and only if R[[x]] is a Baer （PP） ring; （2） If R is an α-rigid ring, then R is a Baer （PP, PS） ring if and only if R[x, x^-1; α] is a Baer （PP, PS） ring.展开更多
Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a...Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.展开更多
In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules.Namely,we introduce the right(resp.,left)n-trivial extension of a category by a family of endofunctors.Among ot...In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules.Namely,we introduce the right(resp.,left)n-trivial extension of a category by a family of endofunctors.Among other results,projective,injective and flat objects of this category are characterized,and two applications are presented at the end of this paper.We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring.展开更多
基金supported by Fondazione Cariverona,Program“Ricerca Scientifica di Eccellenza 2018”(Project“Reducing Complexity in Algebra,Logic,Combinatorics-REDCOM”)supported by China Scholarship Council(Grant No.201906860022)。
文摘Ring epimorphisms often induce silting modules and cosilting modules,termed minimal silting or minimal cosilting.The aim of this paper is twofold.Firstly,we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra.In particular,we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand.Secondly,we discuss the behavior of minimality under ring extensions.We show that minimal cosilting modules over a commutative noetherian ring extend to minimal cosilting modules along any flat ring epimorphism.Similar results are obtained for commutative rings of small homological dimensions.
基金Supported by the National Natural Science Foundation of China(Grant No.11101217)the Natural Science Foundation of Jiangsu Province(Grant No.BK20141476)
文摘Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetric α-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α（1） = 1, then R is a symmetric α-ring if and only if R[x]/（x^n） is a symmetric q^--ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q（R） the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q（R） is a symmetric α-ring. Among others we also show that if a ring R is weakly 2-primal and （α, δ）-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric α^--ring.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1107109711101217)
文摘We introduce the concepts of left （right） zero-divisor rings, a class of rings without identity. We call a ring R left （right） zero-divisor if rR（a） ≠ 0（lR（a） ≠ 0） for every a∈ R, and call R strong left （right） zero-divisor if r R （R）≠0（lR（R）≠ 0）. Camillo and Nielson called a ring right finite annihilated （RFA） if every finite subset has non-zero right annihilator. We present in this paper some basic examples of left zero-divisor rings, and investigate the extensions of strong left zero-divisor rings and RFA rings, giving their equivalent characterizations.
基金Supported by the National Natural Science Foundation of China(Grant No.11071062)the Scientific Research Fundation of Hunan Provincial Education Department(Grant No.12B101)
文摘In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and construct typical examples.We next describe all Σ-associated primes of the Ore extension R[x; α,δ],the skew Laurent polynomial ring R[x,x-1; α] and the skew power series ring R[[x; α]],in terms of the Σ-associated primes of R in a very straightforward way.Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.
基金Supported by grant MTM2007-61673 from the Ministerio de Educación y Ciencia of SpainP06-FQM-01889 from Junta de Andalucía
文摘We focus our attention to the set Gr（ξ） of grouplike elements of a coring ξover a ring A. We do some observations on the actions of the groups U（A） and Aut（ξ~） of units of A and of automorphisms of corings of ξ, respectively, on Gr（ξ）, and on the subset Gal（ξ）of all Galois grouplike elements. Among them, we give conditions on ξ under which Gal（ξ） is a group, in such a way that there is an exact sequence of groups {1} → U（Ag） → U（A） → Gal（ξ） → {1}, where Ag is the subalgebra of coinvaxiants for some g ∈ Gal（ξ）.
基金the Program for New Century Excellent Talents in University(04-0522),and the National Natural Science Foundation of China(10571153)
文摘This paper gives some results on Strong-Armendariz rings and the Ore-extensions R[x,x^-1;α] of Bare, PP and PS rings. And the main two results are： （1） R is a Bear （PP） ring if and only if R[[x]] is a Baer （PP） ring; （2） If R is an α-rigid ring, then R is a Baer （PP, PS） ring if and only if R[x, x^-1; α] is a Baer （PP, PS） ring.
基金Supported by Natural Science Foundation of China (Grant No. A0324656)Natural Science Foundation of Fujian Province (Grant No. 2009J01003)+1 种基金Scientific Research Foundation of Fujian Provincial Department of Science and Technology (Grant No. 2007F5038)Foundation of Fujian Normal University (Grant Nos. 2008100209, 09A004)
文摘Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.
基金Dirar Benkhadra's research reported in this publication was supported by a scholarship from the Graduate Research Assis taut ships in Developing Countries Program of the Commission for Developing Countries of the International Mathematical UnionThe third author was partially supported by the grant MTM2014-54439-P from Ministerio de Economia y Competitividad.
文摘In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules.Namely,we introduce the right(resp.,left)n-trivial extension of a category by a family of endofunctors.Among other results,projective,injective and flat objects of this category are characterized,and two applications are presented at the end of this paper.We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring.
