DC-projective dimensions, Foxby equivalence and SDC-projective modules

2016 ◽  
Vol 15 (06) ◽  
pp. 1650111
Author(s):  
Liang Zhao ◽  
Yiqiang Zhou
Keyword(s):  
Projective Modules ◽  
Frobenius Rings ◽  
Injective Modules ◽  
Semidualizing Module ◽  
Foxby Equivalence

This is a study of Ding projective modules relative to a semidualizing module and related topics. Firstly, we study [Formula: see text]-projective dimensions and [Formula: see text]-projective modules under change of rings. Secondly, we establish a new version of the Foxby equivalence with respect to [Formula: see text]-projective modules and [Formula: see text]-injective modules. Thirdly, we characterize Ding projective modules in [Formula: see text] and Ding injective modules in [Formula: see text]. At last, as applications, some new characterizations of perfect rings and quasi-Frobenius rings are given.

Model structures, recollements and duality pairs

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.

Modules close to the automorphism-invariant and coinvariant

2019 ◽  
Vol 18 (12) ◽  
pp. 1950235 ◽  
Author(s):  
Truong Cong Quynh ◽  
Adel Nailevich Abyzov ◽  
Nguyen Thi Thu Ha ◽  
Tülay Yildirim
Keyword(s):  
General Theory ◽  
General Setting ◽  
Projective Modules ◽  
Injective Modules

The aim of this paper is to introduce a general setting where some well-known results on essentially injective modules, automorphism-(co)invariant modules and small projective modules can be obtained by developing a general theory of modules which are (co)invariant under automorphisms of their covers and envelopes.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
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].

Relative Projective Modules and Relative Injective Modules

2006 ◽  
Vol 34 (7) ◽  
pp. 2403-2418 ◽  
Author(s):  
Lixin Mao ◽  
Nanqing Ding
Keyword(s):  
Projective Modules ◽  
Injective Modules

Graded injective modules and graded quasi-Frobenius rings

2015 ◽  
Vol 9 ◽  
pp. 1113-1123
Author(s):  
Salah El Din S. Hussien ◽  
Essam El-Seidy ◽  
Manar E. Tabarak
Keyword(s):  
Frobenius Rings ◽  
Injective Modules

scholarly journals The New Results in n-injective Modules and n-projective Modules

2021 ◽  
pp. 271-285
Author(s):  
Samira Hashemi ◽  
Feysal Hassani ◽  
Rasul Rasuli
Keyword(s):  
Exact Sequence ◽  
Projective Module ◽  
Injective Module ◽  
Projective Modules ◽  
Injective Modules

In this paper, we introduce and clarify a new presentation between the n-exact sequence and the n-injective module and n-projective module. Also, we obtain some new results about them.

On relative counterpart of Auslander’s conditions

2021 ◽  
Author(s):  
Driss Bennis ◽  
Rachid El Maaouy ◽  
J. R. García Rozas ◽  
Luis Oyonarte
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
New Approach ◽  
Gorenstein Rings ◽  
Semidualizing Module ◽  
Gorenstein Projective Modules ◽  
Frobenius Category ◽  
Relative Notion ◽  
Frobenius Categories

It is now well known that the conditions used by Auslander to define the Gorenstein projective modules on Noetherian rings are independent. Recently, Ringel and Zhang adopted a new approach in investigating Auslander’s conditions. Instead of looking for examples, they investigated rings on which certain implications between Auslander’s conditions hold. In this paper, we investigate the relative counterpart of Auslander’s conditions. So, we extend Ringel and Zhang’s work and introduce other concepts. Namely, for a semidualizing module [Formula: see text], we introduce weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings as rings representing relations between the relative counterpart of Auslander’s conditions. Moreover, we introduce a relative notion of the well-known Frobenius category. We show how useful are [Formula: see text]-Frobenius categories in characterizing weakly [Formula: see text]-Gorenstein and partially [Formula: see text]-Gorenstein rings.

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