scholarly journals Gorenstein flat precovers and Gorenstein injective preenvelopes in Grothendieck categories

2019 ◽  
Vol 57 (1) ◽  
pp. 55-83
Author(s):  
Edgar Enochs ◽  
J.R. García Rozas ◽  
Luis Oyonarte ◽  
Blas Torrecillas
Keyword(s):  
Gorenstein Flat ◽  
Gorenstein Injective

Gorenstein Injective and Gorenstein Flat Dimensions Under Base Change

2003 ◽  
Vol 31 (2) ◽  
pp. 991-1005 ◽  
Author(s):  
Leila Khatami ◽  
Siamak Yassemi
Keyword(s):  
Base Change ◽  
Gorenstein Flat ◽  
Gorenstein Injective

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).

scholarly journals GORENSTEIN PROJECTIVE, INJECTIVE AND FLAT MODULES

2009 ◽  
Vol 87 (3) ◽  
pp. 395-407 ◽  
Author(s):  
ZHONGKUI LIU ◽  
XIAOYAN YANG
Keyword(s):  
Homological Algebra ◽  
Flat Modules ◽  
Gorenstein Flat ◽  
Gorenstein Injective

AbstractIn basic homological algebra, projective, injective and flat modules play an important and fundamental role. In this paper, we discuss some properties of Gorenstein projective, injective and flat modules and study some connections between Gorenstein injective and Gorenstein flat modules. We also investigate some connections between Gorenstein projective, injective and flat modules under change of rings.

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

2004 ◽  
Vol 32 (11) ◽  
pp. 4415-4432
Author(s):  
Javad Asadollahi ◽  
Shokrollah Salarian
Keyword(s):  
Gorenstein Rings ◽  
Gorenstein Flat ◽  
Gorenstein Injective

scholarly journals STABILITY OF GORENSTEIN FLAT CATEGORIES

2011 ◽  
Vol 54 (1) ◽  
pp. 177-191 ◽  
Author(s):  
GANG YANG ◽  
ZHONGKUI LIU
Keyword(s):  
Exact Sequence ◽  
Gorenstein Flat ◽  
Gorenstein Injective

AbstractA left R-module M is called two-degree Gorenstein flat if there exists an exact sequence of Gorenstein flat left R-modules ⋅⋅⋅ → G2 → G1 → G0 → G−1 → G−2 → ⋅⋅⋅ such that M ≅ Ker(G0 → G−1) and it remains exact after applying H ⊗R- for any Gorenstein injective right R-module H. In this paper we first give some characterisations of Gorenstein flat objects in the category of complexes of modules and then use them to show that two notions of the two-degree Gorenstein flat and the Gorenstein flat left R-modules coincide when R is right coherent.

Ω-Gorenstein Projective, Injective and Flat Modules

2011 ◽  
Vol 18 (02) ◽  
pp. 273-288
Author(s):  
Xiaoyan Yang ◽  
Zhongkui Liu
Keyword(s):  
Flat Modules ◽  
Gorenstein Flat ◽  
Gorenstein Injective

In this paper, we study some properties of Ω-Gorenstein projective, injective and flat modules and discuss some connections between Ω-Gorenstein injective and Ω-Gorenstein flat modules. We also consider these properties under change of rings.

scholarly journals STABILITY OF GORENSTEIN FLAT MODULES

2011 ◽  
Vol 54 (1) ◽  
pp. 169-175 ◽  
Author(s):  
SAMIR BOUCHIBA ◽  
MOSTAFA KHALOUI
Keyword(s):  
Projective Modules ◽  
Natural Question ◽  
Flat Module ◽  
Flat Modules ◽  
Closed Ring ◽  
Gorenstein Projective Modules ◽  
Gorenstein Categories ◽  
Gorenstein Flat ◽  
Gorenstein Injective ◽  
Gorenstein Flat Module

AbstractSather-Wagstaff et al. proved in [8] (S. Sather-Wagsta, T. Sharif and D. White, Stability of Gorenstein categories, J. Lond. Math. Soc.(2), 77(2) (2008), 481–502) that iterating the process used to define Gorenstein projective modules exactly leads to the Gorenstein projective modules. Also, they established in [9] (S. Sather-Wagsta, T. Sharif and D. White, AB-contexts and stability for Goren-stein at modules with respect to semi-dualizing modules, Algebra Represent. Theory14(3) (2011), 403–428) a stability of the subcategory of Gorenstein flat modules under a procedure to build R-modules from complete resolutions. In this paper we are concerned with another kind of stability of the class of Gorenstein flat modules via-à-vis the very Gorenstein process used to define Gorenstein flat modules. We settle in affirmative the following natural question in the setting of a left GF-closed ring R: Given an exact sequence of Gorenstein flat R-modules G = ⋅⋅⋅ G2G1G0G−1G−2 ⋅⋅⋅ such that the complex H ⊗RG is exact for each Gorenstein injective right R-module H, is the module M:= Im(G0 → G−1) a Gorenstein flat module?

Noncommutative G-hereditary rings

2020 ◽  
pp. 2150052
Author(s):  
Yunxia Li ◽  
Jian Wang ◽  
Yuxian Geng ◽  
Jiangsheng Hu
Keyword(s):  
Commutative Ring ◽  
Associative Ring ◽  
Partial Answer ◽  
Hereditary Ring ◽  
Gorenstein Flat ◽  
Gorenstein Injective

In this paper, we introduce and study left (right) [Formula: see text]-hereditary rings over any associative ring, and these rings are exactly [Formula: see text]-hereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. As applications, we give a partial answer to the question posed by Bazzoni, Cortés-Izurdiaga and Estrada, and characterize when the character module of a Gorenstein injective left [Formula: see text]-module is Gorenstein flat provided that [Formula: see text] is a left [Formula: see text]-hereditary ring. In addition, some new characterizations of left hereditary rings are given.

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