Remarks on Gorenstein Weak Injective and Weak Flat Modules

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.

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Acyclic Complexes and Gorenstein Rings

Algebra Colloquium ◽

10.1142/s1005386720000474 ◽

2020 ◽

Vol 27 (03) ◽

pp. 575-586

Author(s):

Sergio Estrada ◽

Alina Iacob ◽

Holly Zolt

Keyword(s):

Analogous Result ◽

Gorenstein Ring ◽

Gorenstein Rings ◽

Injective Modules ◽

Flat Modules ◽

Coherent Ring ◽

Acyclic Complexes ◽

Gorenstein Flat ◽

Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-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 [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary 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-module 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 M+ 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).

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Model structures, recollements and duality pairs

Journal of Algebra and Its Applications ◽

10.1142/s0219498823500172 ◽

2021 ◽

Author(s):

Wenjing Chen ◽

Zhongkui Liu

Keyword(s):

Projective Modules ◽

Cotorsion Pair ◽

Gorenstein Rings ◽

Injective Modules ◽

Flat Modules ◽

Duality Pairs ◽

Cotorsion Pairs ◽

Gorenstein Flat ◽

Model Structures

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.

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Strict Mittag–Leffler modules over Gorenstein injective modules

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500504 ◽

2019 ◽

Vol 19 (03) ◽

pp. 2050050 ◽

Cited By ~ 1

Author(s):

Yanjiong Yang ◽

Xiaoguang Yan

Keyword(s):

Noetherian Ring ◽

Direct Limit ◽

Injective Module ◽

Noetherian Rings ◽

Projective Modules ◽

Injective Modules ◽

Pure Submodules ◽

Covering Class ◽

Gorenstein Injective ◽

Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

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Canonical filtrations of Gorenstein injective modules

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-2010-10686-x ◽

2010 ◽

Vol 139 (7) ◽

pp. 2415-2421 ◽

Cited By ~ 1

Author(s):

Edgar E. Enochs ◽

Zhaoyong Huang

Keyword(s):

Injective Modules ◽

Gorenstein Injective

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On gorenstein injective modules

Communications in Algebra ◽

10.1080/00927879308824744 ◽

1993 ◽

Vol 21 (10) ◽

pp. 3489-3501 ◽

Cited By ~ 34

Author(s):

Edgar E. Enochs ◽

Overtoun M.G. Jenda

Keyword(s):

Injective Modules ◽

Gorenstein Injective

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WEAK AB-CONTEXT FOR FP-INJECTIVE MODULES WITH RESPECT TO SEMIDUALIZING MODULES

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500394 ◽

2013 ◽

Vol 12 (07) ◽

pp. 1350039 ◽

Cited By ~ 2

Author(s):

JIANGSHENG HU ◽

DONGDONG ZHANG

Keyword(s):

Noetherian Ring ◽

Model Structure ◽

Injective Dimension ◽

Commutative Noetherian Ring ◽

New Model ◽

Injective Modules ◽

Semidualizing Modules ◽

Semidualizing Bimodule ◽

Gorenstein Injective

Let S and R be rings and SCR a semidualizing bimodule. We define and study GC-FP-injective R-modules, and these modules are exactly C-Gorenstein injective R-modules defined by Holm and Jørgensen provided that S = R is a commutative Noetherian ring. We mainly prove that the category of GC-FP-injective R-modules is a part of a weak AB-context, which is dual of weak AB-context in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander–Buchweitz approximations for R-modules with finite GC-FP-injective dimension. As an application, a new model structure in Mod R is given.

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