A generalization of Gorenstein injective modules

2021 ◽  
Vol 14 (3) ◽  
Author(s):  
Fatemeh Mohammadi Aghjeh Mashhad
Keyword(s):  
Injective Modules ◽  
Gorenstein Injective

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].

On gorenstein injective modules

1993 ◽  
Vol 21 (10) ◽  
pp. 3489-3501 ◽  
Author(s):  
Edgar E. Enochs ◽  
Overtoun M.G. Jenda
Keyword(s):  
Injective Modules ◽  
Gorenstein Injective

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

2013 ◽  
Vol 12 (07) ◽  
pp. 1350039 ◽  
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.

scholarly journals Acyclic Complexes and Gorenstein Rings

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).

Weak Gorenstein Injective Modules

2021 ◽  
Vol 11 (05) ◽  
pp. 922-928
Author(s):  
珍珍 朵
Keyword(s):  
Injective Modules ◽  
Gorenstein Injective

ω-Gorenstein injective modules and dimensions

2010 ◽  
Vol 54 (3) ◽  
pp. 421-436 ◽  
Author(s):  
HaiYan Zhu ◽  
NanQing Ding
Keyword(s):  
Injective Modules ◽  
Gorenstein Injective
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