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n-strongly Gorenstein Projective and Injective and Flat Modules 被引量:3
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作者 YANG Xiao-yan 《Chinese Quarterly Journal of Mathematics》 CSCD 2014年第4期553-564,共12页
In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applicati... In this paper, we study some properties of n-strongly Gorenstein projective,injective and flat modules, and discuss some connections between n-strongly Gorenstein injective, projective and flat modules. Some applications are given. 展开更多
关键词 n-strongly gorenstein projective module n-strongly gorenstein injective module n-strongly gorenstein flat module
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On Rings with Finite Global Gorenstein Dimensions
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作者 REN Wei ZHANG Yu 《Chinese Quarterly Journal of Mathematics》 2017年第3期255-260,共6页
As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generali... As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou's equality of global Gorenstein dimension is given. 展开更多
关键词 generalized gorenstein ring gorenstein projective(injective) dimension
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Gorenstein flatness and injectivity over Gorenstein rings 被引量:2
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作者 SONG WeiLing HUANG ZhaoYong 《Science China Mathematics》 SCIE 2008年第2期215-218,共4页
Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left ... Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results. 展开更多
关键词 gorenstein flat gorenstein injective gorenstein rings 16E10 16E30
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Remarks on Gorenstein Weak Injective and Weak Flat Modules 被引量:6
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作者 Tiwei Zhao Yunge Xu 《Algebra Colloquium》 SCIE CSCD 2020年第4期687-702,共16页
In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than t... In this paper,we introduce Gorenstein weak injective and weak flat modules in terms of,respectively,weak injective and weak flat modules;the classes of Gorenstein weak injective and weak flat modules are larger than the classical classes of Gorenstein injective and flat modules.In this new setting,we characterize rings over which all modules are Gorenstein weak injective.Moreover,we discuss the relation between the weak cosyzygy and Gorenstein weak cosyzygy of a module,and also the stability of Gorenstein weak injective modules. 展开更多
关键词 weak injective module weak flat module gorenstein weak injective module gorenstein weak flat module cosyzygy
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Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis) 被引量:5
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作者 CHRISTENSEN Lars Winthe KOKSAL Fatih LIANG Li 《Science China Mathematics》 SCIE CSCD 2017年第3期401-420,共20页
For a commutative ring R and a faithfully fiat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S R M is Gorenstein flat, and that an R-module ... For a commutative ring R and a faithfully fiat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S R M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Homn(S, N) is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change. 展开更多
关键词 gorenstein injective dimension faithfully flat co-base change Corenstein flat dimension faithfullyflat base change
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A Note on DG-Gorenstein Injective Complexes 被引量:4
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作者 Bo Lu Kaiyang Lan 《Algebra Colloquium》 SCIE CSCD 2020年第4期731-740,共10页
The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand ... The notion of DG-Gorenstein injective complexes is studied in this article.It is shown that a complex G is DG-Gorenstein injective if and only if G is exact with Z_(n)(G)Gorenstein injective in R-Mod for each n∈Zand any morphism f:E→G is null homotopic whenever E is a DG-injective complex. 展开更多
关键词 gorenstein injective module DG-injective complex DG-gorenstein injective complex
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Weak Gorenstein Cotorsion Modules 被引量:2
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作者 Liang Zhao Jiaqun Wei Jiangsheng Hu 《Algebra Colloquium》 SCIE CSCD 2018年第2期265-276,共12页
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
关键词 weak gorenstein cotorsion module weak gorenstein cotorsion injective dimension global weak Corenstein cotorsion dimension
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Acyclic Complexes and Gorenstein Rings
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作者 Sergio Estrada Alina Iacob Holly Zolt 《Algebra Colloquium》 SCIE CSCD 2020年第3期575-586,共12页
For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that... For a given class of modules A,let A be the class of exact complexes having all cycles in A,and dw(A)the class of complexes with all components in A.Denote by GL the class of Gorenstein injective modules.We prove that the following are equivalent over any ring R:every exact complex of injective modules is totally acyclic;every exact complex of Gorenstein injective modules is in every complex in dw(GL)is dg-Gorenstein injective.The analogous result for complexes of flat and Gorenstein flat modules also holds over arb计rary rings.If the ring is n-perfect for some integer n≥0,the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules.We also prove the following characterization of Gorenstein rings.Let R be a commutative coherent ring;then the following are equivalent:(1)every exact complex of FP-injective modules has all its cycles Ding injective modules;(2)every exact complex of flat modules is F-totally acyclic,and every R-modulc M such that M^(+)is Gorenstein flat is Ding injective;(3)every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that is Gorenstein flat is Ding injective.If R has finite Krull dimension,statements(1)-(3)are equivalent to(4)R is a Gorenstein ring(in the sense of Iwanaga). 展开更多
关键词 totally acyclic complex gorenstein injective module gorenstein projective module gorenstein flat module Ding injective module
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Gorenstein Injective and Injective Complete Cohomological Dimensions of Groups
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作者 Abdolnaser Bahlekeh 《Algebra Colloquium》 SCIE CSCD 2015年第3期469-478,共10页
Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group P, called the injective complete cohomological dimension of F, denoted by iccd P. We study this dimension and ... Using Nucinkis's injective complete cohomological functors, we assign a numerical invariant to each group P, called the injective complete cohomological dimension of F, denoted by iccd P. We study this dimension and investigate its properties. Also, we define the Gorenstein injective dimension of the group F, which is denoted by Gid F. We show that Gid F is related to iccd F, as well as to spli and silp invariants of Gedrich and Gruenberg. In particular, it is shown that iccd P is a refinement of Gid P. In addition, we show that silp F = spli F 〈 ∞if and only if the Shapiro lemma holds for injective complete cohomology. 展开更多
关键词 I-complete cohomology complete injective resolution gorenstein injective modules
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On Gorenstein coalgebras
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作者 Xuerong FU Hailou YAO 《Frontiers of Mathematics in China》 SCIE CSCD 2016年第4期845-867,共23页
We investigate the comodule representation category over the Morita-Takeuchi context coalgebra F and study the Gorensteinness of F. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-... We investigate the comodule representation category over the Morita-Takeuchi context coalgebra F and study the Gorensteinness of F. Moreover, we determine explicitly all Gorenstein injective comodules over the Morita-Takeuchi context coalgebra F and discuss the localization in Gorenstein coalgebras. In particular, we describe its Gabriel quiver and carry out some examples when the Morita-Takeuchi context coalgebra is basic. 展开更多
关键词 gorenstein coalgebra gorenstein injective comodule Gabriel quiver
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Gorenstein injective sheaves
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作者 Sergio ESTRADA Alina IACOB 《Frontiers of Mathematics in China》 SCIE CSCD 2017年第1期87-95,共9页
We define Gorenstein injective quasi-coherent sheaves, and prove that the notion is local in case the scheme is Gorenstein. We also give a new characterization of a Gorenstein scheme in terms of the total acyclicity o... We define Gorenstein injective quasi-coherent sheaves, and prove that the notion is local in case the scheme is Gorenstein. We also give a new characterization of a Gorenstein scheme in terms of the total acyclicity of every acyclic complex of injective quasi-coherent modules. 展开更多
关键词 gorenstein injective Noetherian scheme
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Some Remarks on Projective Generators and Injective Cogenerators 被引量:12
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作者 Li LIANG Nan Qing DING Gang YANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2014年第12期2063-2078,共16页
In this paper,we give a relationship between projective generators(resp.,injective cogenerators) in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get th... In this paper,we give a relationship between projective generators(resp.,injective cogenerators) in the category of R-modules and the counterparts in the category of complexes of R-modules.As a consequence,we get that complexes of W^--Gorenstein modules are actually W-Gorenstein complexes whenever W is a subcategory of R-modules satisfying W⊥W,where W^- is the subcategory of exact complexes with all cycles in W.We also study when all cycles of a W-Gorenstein complexes are W^--Gorenstein modules. 展开更多
关键词 W-gorenstein modules gorenstein projective complexes gorenstein injective complexes projective generators injective cogenerators
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