Let R be a ring. We define a dimension, called P-cotorsion dimension, for modules and rings. The aim of this article is to investigate P-cotorsion dimensions of modules and rings and the relations between P-cotorsion ...Let R be a ring. We define a dimension, called P-cotorsion dimension, for modules and rings. The aim of this article is to investigate P-cotorsion dimensions of modules and rings and the relations between P-cotorsion dimension and other homological dimensions. This dimension has nice properties when the ring in consideration is generalized morphic.展开更多
Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cot...Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.展开更多
In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. W...In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd（M） = IFP-id（M＋） and IFP-fd（M＋）=IFP-id（M） for any R-module M over any ring R. Let ：Z-In （resp., ＂Zgv,~） he the class of all left （resp., right） R-modules of IFP-injective （resp., IFP-flat） dimension at most n. We prove that every right R-module has an IFn- preenvelope, （IFn,IF⊥n） is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id（RR） 〈 n, （IIn,II⊥n） is a perfect cotorsion theory. This generalizes and improves the earlier work （J. Algebra 242 （2001）, 447-459）. Finally, some applications are given.展开更多
In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a co...In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.展开更多
Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under...Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under the assumption that R^σ G is a,separable extension over R^σ H,it is shown that R^σ G and R^σ H share the same homological dimensions.Several known results are then obtained as corollaries.Moreover,we investigate the relationships between the homological dimensions of Ra G and the homological dimensions of a commutative ring R,using the trivial R^σ G-module.展开更多
Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modu...Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.展开更多
基金supported by Collegial Natural Science Research Program of Education Department of Jiangsu Province (07KJD110043)
文摘Let R be a ring. We define a dimension, called P-cotorsion dimension, for modules and rings. The aim of this article is to investigate P-cotorsion dimensions of modules and rings and the relations between P-cotorsion dimension and other homological dimensions. This dimension has nice properties when the ring in consideration is generalized morphic.
基金The NSF(KJ2016A545,KJ2015B12,2017ZR08zd)of Anhui Provincethe key projectsoutstanding young talent support program(gxyq ZD2016353)of Anhui Province
文摘Let H be a finite-dimensional weak Hopf algebra over a field κ and A an associative algebra, and A#;H a weak crossed product. In this paper, a spectral sequence for Ext is constructed which yields an estimate for cotorsion dimension of A#;H in terms of the corresponding data for H and A.
基金supported by National Natural Science Foundation of China(10961021,11001222)
文摘In basic homological algebra, the flat and injective dimensions of modules play an important and fundamental role. In this paper, the closely related IFP-flat and IFP-injective dimensions are introduced and studied. We show that IFP-fd（M） = IFP-id（M＋） and IFP-fd（M＋）=IFP-id（M） for any R-module M over any ring R. Let ：Z-In （resp., ＂Zgv,~） he the class of all left （resp., right） R-modules of IFP-injective （resp., IFP-flat） dimension at most n. We prove that every right R-module has an IFn- preenvelope, （IFn,IF⊥n） is a perfect cotorsion theory over any ring R, and for any ring R with IFP-id（RR） 〈 n, （IIn,II⊥n） is a perfect cotorsion theory. This generalizes and improves the earlier work （J. Algebra 242 （2001）, 447-459）. Finally, some applications are given.
基金Supported by the National Natural Science Foundation of China(11201424)the Zhejiang Natural Science Foundation of China(LY12A01026)
文摘In this paper, we investigate Ding projective dimensions and Ding injective di- mensions of modules and rings. Let R be a ring with rDPD（R） = n 〈 ∞, and let YYl = {Mild（M） 〈 ∞}. We prove that （DP,W1） is a complete hereditary cotorsion pair such that a module M belongs to DD∩W1 if and only if M is projective, moreover, 1421 = （M[pd（M） 〈 ∞} = {MIfd（M） ≤ n} = {MIpd（M） ≤ n}. Then we introduce and inves- tigate Ding derived functor Dext^i（-, -）, and use it to characterize global Ding dimension. We show that if R is a Ding-Chen ring, or if R is a ring with rDPD（R） 〈≤ and rDID（R） 〈 ≤, then rDPD（R） 〈 n if and only if rDID（R） 〈 n if and only if Dext^n＋i（M,N） = 0 for all modules M and N and all integer i ≥ 1.
基金supported by the Scientific Research Foundation of Hunan Provincial Education Department(no.18C0997).
文摘Let R be a ring and let H be a.subgroup of a finite group G.We consider the weak global dimension,cotorsion dimension and weak Gorenstein global dimension of the skew group ring R^σ G and its coefficient ring R.Under the assumption that R^σ G is a,separable extension over R^σ H,it is shown that R^σ G and R^σ H share the same homological dimensions.Several known results are then obtained as corollaries.Moreover,we investigate the relationships between the homological dimensions of Ra G and the homological dimensions of a commutative ring R,using the trivial R^σ G-module.
基金The authors wish to express their sincere thanks to the referees for their valuable comments and suggestions. The first author was supported by the Postdoctoral Science Foundation of China （2017M611851）, the Jiangsu Planned Projects for Postdoctoral Research Funds （1601151C） and the Provincial Natural Science Foundation of Anhui Province of China （KJ2017A040）. The second author was supported by the NSFC （11771212）, and the first two authors were supported by a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The third author was supported by the NSFC （11501257, 11671069, 11771212） and the Postdoctoral Science Foundation of China （2016M600426）.
文摘Let R be a ring and let be the class of strongly Gorenstein fiat right R-modules. We call a right R-module M a weak Gorenstein cotorsion module if M is in the class ⊥. Properties of weak Gorenstein cotorsion modules are investigated. It is shown that weak Gorenstein cotorsion R-modules over coherent rings are indeed weaker than Gorenstein cotorsion R-modules. Weak Gorenstein cotorsion dimension for modules and rings are also studied.
