GORENSTEIN FP-INJECTIVE AND GORENSTEIN FLAT MODULES

2008 ◽  
Vol 07 (04) ◽  
pp. 491-506 ◽  
Author(s):  
LIXIN MAO ◽  
NANQING DING
Keyword(s):  
Exact Sequence ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Rings ◽  
Gorenstein Flat

In this paper, Gorenstein FP-injective modules are introduced and studied. An R-module M is called Gorenstein FP-injective if there is an exact sequence ⋯ → E1 → E0 → E0 → E1 → ⋯ of injective R-modules with M = ker (E0 → E1) and such that Hom (E, -) leaves the sequence exact whenever E is an FP-injective R-module. Some properties of Gorenstein FP-injective and Gorenstein flat modules over coherent rings are obtained. Several known results are extended.

scholarly journals STRONGLY GORENSTEIN FLAT MODULES

2009 ◽  
Vol 86 (3) ◽  
pp. 323-338 ◽  
Author(s):  
NANQING DING ◽  
YUANLIN LI ◽  
LIXIN MAO
Keyword(s):  
Exact Sequence ◽  
Projective Modules ◽  
Flat Dimension ◽  
Flat Modules ◽  
Coherent Rings ◽  
Gorenstein Flat ◽  
Strongly Gorenstein Flat Modules

AbstractIn this paper, strongly Gorenstein flat modules are introduced and investigated. An R-module M is called strongly Gorenstein flat if there is an exact sequence ⋯→P1→P0→P0→P1→⋯ of projective R-modules with M=ker (P0→P1) such that Hom(−,F) leaves the sequence exact whenever F is a flat R-module. Several well-known classes of rings are characterized in terms of strongly Gorenstein flat modules. Some examples are given to show that strongly Gorenstein flat modules over coherent rings lie strictly between projective modules and Gorenstein flat modules. The strongly Gorenstein flat dimension and the existence of strongly Gorenstein flat precovers and pre-envelopes are also studied.

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.

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).

Some remarks on Nonnil-coherent rings and ϕ-IF rings

2021 ◽  
pp. 2250211
Author(s):  
Wei Qi ◽  
Xiaolei Zhang
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Injective Module ◽  
Nilpotent Radical ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Rings

Let [Formula: see text] be a commutative ring. If the nilpotent radical [Formula: see text] of [Formula: see text] is a divided prime ideal, then [Formula: see text] is called a [Formula: see text]-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and [Formula: see text]-coherent rings introduced by Bacem and Ali [Nonnil-coherent rings, Beitr. Algebra Geom. 57(2) (2016) 297–305], and then characterize nonnil-coherent rings in terms of [Formula: see text]-flat modules, nonnil-injective modules and nonnil-FP-injective modules. A [Formula: see text]-ring [Formula: see text] is called a [Formula: see text]-IF ring if any nonnil-injective module is [Formula: see text]-flat. We obtain some module-theoretic characterizations of [Formula: see text]-IF rings. Two examples are given to distinguish [Formula: see text]-IF rings and IF [Formula: see text]-rings.

PERIODIC SHORT EXACT SEQUENCES AND PERIODIC PURE-EXACT SEQUENCES

2010 ◽  
Vol 09 (06) ◽  
pp. 859-870 ◽  
Author(s):  
SAMIR BOUCHIBA ◽  
MOSTAFA KHALOUI
Keyword(s):  
Exact Sequence ◽  
Finite Groups ◽  
Projective Module ◽  
Homological Algebra ◽  
Flat Module ◽  
Injective Modules ◽  
Flat Modules ◽  
Gorenstein Homological Algebra ◽  
Exact Sequences

Benson and Goodearl [Periodic flat modules, and flat modules for finite groups, Pacific J. Math.196(1) (2000) 45–67] proved that if M is a flat module over a ring R such that there exists an exact sequence of R-modules 0 → M → P → M → 0 with P a projective module, then M is projective. The main purpose of this paper is to generalize this theorem to any exact sequence of the form 0 → M → G → M → 0, where G is an arbitrary module over R. Moreover, we seek counterpart entities in the Gorenstein homological algebra of pure projective and pure injective modules.

FPn-Injective, FPn-Flat Covers and Preenvelopes, and Gorenstein AC-Flat Covers

2018 ◽  
Vol 25 (02) ◽  
pp. 319-334
Author(s):  
Daniel Bravo ◽  
Sergio Estrada ◽  
Alina Iacob
Keyword(s):  
Direct Summand ◽  
Flat Module ◽  
Injective Modules ◽  
Flat Modules ◽  
Gorenstein Flat ◽  
Strongly Gorenstein Flat Modules

We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of FP∞-injective and FP∞-flat modules (i.e., absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules, i.e., the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example, we show that this class is precovering over any ring R. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. Moreover, we prove that if the class of Gorenstein AC-flat modules is closed under extensions, then it is covering.

Noncommutative G-semihereditary rings

2018 ◽  
Vol 17 (01) ◽  
pp. 1850014 ◽  
Author(s):  
Jian Wang ◽  
Yunxia Li ◽  
Jiangsheng Hu
Keyword(s):  
Associative Ring ◽  
Sufficient Condition ◽  
Projective Modules ◽  
The Third ◽  
Flat Modules ◽  
Gorenstein Projective Modules ◽  
Direct Limits ◽  
Gorenstein Flat ◽  
Finitely Presented ◽  
Semihereditary Rings

In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.

Notes on Gorenstein Flat Modules

2018 ◽  
Vol 45 (2) ◽  
pp. 337-344
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Flat Modules ◽  
Gorenstein Flat

Relative FP-gr-injective and gr-flat modules

2018 ◽  
Vol 28 (06) ◽  
pp. 959-977 ◽  
Author(s):  
Tiwei Zhao ◽  
Zenghui Gao ◽  
Zhaoyong Huang
Keyword(s):  
Graded Modules ◽  
Flat Modules ◽  
Duality Pairs ◽  
Coherent Rings ◽  
Flat Cover ◽  
Finitely Presented

Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope.

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