Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a ...Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is introduced and studied.展开更多
Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategor...Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategory of A.Then the right 1-orthogonal category G(C)^⊥1 of G(C)is both projectively resolving and injectively coresolving in A.We also get that the subcategory SPC(G(C))of A consisting of objects admitting special G(C)-precovers is closed under extensions and C-stable direct summands(*).Furthermore,if C is a generator for G(C)^⊥1,then we have that SPC(G(C))is the minimal subcategory of A containing G(C)^⊥1∪G(C)with respect to the property(*),and that SPC(G(C))is C-resolving in A with a C-proper generator C.展开更多
Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the propert...Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.展开更多
This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows th...This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤展开更多
In this paper,we consider some generalizations of tilting torsion classes and cotilting torsion-free classes,give the definition and characterizations of n-tilting torsion classes and n-cotilting torsion-free classes,...In this paper,we consider some generalizations of tilting torsion classes and cotilting torsion-free classes,give the definition and characterizations of n-tilting torsion classes and n-cotilting torsion-free classes,and study n-tilting preenvelopes and n-cotilting precovers.展开更多
基金This research was partially supported by the Specialized Research Fund for the Doctoral Program of Higher ducation of China (20020284009, 20030284033) NSF of China (10331030) Jiangsu Planned Projects for Postdoctoral Research Funds (0203003403)the Postdoctoral Research Funds of China (2006037713)and the Nanjing Institute of Technology of China.
文摘Let C be a class of R-modules closed under isomorphisms and finite direct sums. It is first shown that the finite direct sum of almost C-precovers is an almost C-precover and the direct sum of an almost C-cover and a weak C-cover is a weak C-cover. Then the notion of almost C-preenvelopes is introduced and studied.
基金supported by National Natural Science Foundation of China (Grant No. 11571164)Priority Academic Program Development of Jiangsu Higher Education Institutions+1 种基金the University Postgraduate Research and Innovation Project of Jiangsu Province 2016 (Grant No. KYZZ16 0034)Nanjing University Innovation and Creative Program for PhD Candidate (Grant No. 2016011)
文摘Let A be an abelian category and P(A)be the subcategory of A consisting of projective objects.Let C be a full,additive and self-orthogonal subcategory of A with P(A)a generator,and let G(C)be the Gorenstein subcategory of A.Then the right 1-orthogonal category G(C)^⊥1 of G(C)is both projectively resolving and injectively coresolving in A.We also get that the subcategory SPC(G(C))of A consisting of objects admitting special G(C)-precovers is closed under extensions and C-stable direct summands(*).Furthermore,if C is a generator for G(C)^⊥1,then we have that SPC(G(C))is the minimal subcategory of A containing G(C)^⊥1∪G(C)with respect to the property(*),and that SPC(G(C))is C-resolving in A with a C-proper generator C.
基金supported by National Natural Science Foundation of China (Grant Nos. 11671069 and 11771212)Zhejiang Provincial Natural Science Foundation of China (Grant No. LY18A010032)+1 种基金Qing Lan Project of Jiangsu Province and Jiangsu Government Scholarship for Overseas Studies (Grant No. JS2019-328)during a visit of the first author to Charles University in Prague with the support by Jiangsu Government Scholarship
文摘Let T be a right exact functor from an abelian category B into another abelian category A.Then there exists a functor p from the product category A×B to the comma category(T↓A).In this paper,we study the property of the extension closure of some classes of objects in(T↓A),the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X,Y)in(T↓A).Moreover,we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in(T↓A).As an application,we prove that under suitable conditions,the class of Gorenstein projective leftΛ-modules over a triangular matrix ringΛ=(R M 0 S)is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering.Consequently,we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
基金the National Natural Science Foundation of China (No.10171082) the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of the Ministry of Education of China and NWNU-KJCXGC212.
文摘This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤
基金Supported by the 2018 Scientific Research Projects in Universities of Gansu Province(2018A-269)
文摘In this paper,we consider some generalizations of tilting torsion classes and cotilting torsion-free classes,give the definition and characterizations of n-tilting torsion classes and n-cotilting torsion-free classes,and study n-tilting preenvelopes and n-cotilting precovers.
