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
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
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
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 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.
