Stability of Gorenstein Flat Modules

Bouchiba, Samir; Khaloui, Mostafa

doi:10.1017/s0017089511000516

Cited by 15 publications

(12 citation statements)

References 8 publications

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“…ACKNOWLEDGEMENTS. We are informed by the Editor-in-Chief of the Glasgow Mathematical Journal that an independent proof of this result has been obtained by Samir Bouchiba and Mostafa Khaloui [4]. The authors thank the referee for his/her careful reading and many considerable suggestions, which have improved this paper.…”

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confidence: 82%

Stability of Gorenstein Flat Categories

Yang

Liu

2011

Glasgow Math. J.

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A 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.

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confidence: 82%

Stability of Gorenstein Flat Categories

Yang

Liu

2011

Glasgow Math. J.

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“…For more details, see [1]. The stabiltity of Gorenstein flat R-module has been treated by Bouchiba and Khaloui [2], Xu and Ding [13], Yang and Liu [17], respectively. By using totally different techniques, they showed that over a left GF-closed ring R (a ring R over which the class of the Gorenstein flat R-modules is closed under extensions), an R-module M is Gorenstein flat if and only if there exists an exact sequence of Gorenstein flat (1) M ∈ G(A).…”

Section: Stability Of Gorenstein Categories With Respect To Cotorsionmentioning

confidence: 99%

Gorenstein projective complexes with respect to cotorsion pairs

Zhao¹,

Ma²

2018

Czech.Math.J.

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Let (A, B) be a complete and hereditary cotorsion pair in the category of left R-modules. In this paper, the so-called Gorenstein projective complexes respect to the cotorsion pair (A, B) are introduced. We show that these complexes are just the complexes of Gorenstein projective modules respect to the cotorsion pair (A, B). As applications, we prove that both Gorenstein projective modules with respect to cotorsion pairs and Gorenstein projective complexes with respect to cotorsion pairs possess of stability.Keywords: cotorsion pair, Gorenstein projective complex with respect to cotorsion pairs, stability. MSC 2010: 18G25, 18G35This result unifies and generalizes of [10, Theorems 4.7] and [18, Theorems 2.2, 3.1]. By this connection, we prove the following results (see Theorem 4.1 and Theorem 4.2, respectively). Theorem 1.2. A left R-module M belongs to G(A) if and only if there exists a Hom R (−, A ∩ B)-exact exact sequence · · ·

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“…Since Tor R j≥1 (I C (R), A i ) = 0 for any i ≥ 1, we get an exact sequence 0−→ N 0 −→ G ′ −→ G 2 −→ G 3 −→ • • • in M(R), which is I C (R)⊗ R − exact.Repeating the process, we obtain the desired exact sequence. Thus, A ∈ GF C (R).In the special case C = R, we obtain the main theorem of[2] and[17, Theorem 4.3]. Corollary 3.13.…”

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confidence: 91%

Resolutions and stability of $C$-Gorenstein flat modules

Zhao¹,

Xia²

2016

Rocky Mountain J. Math.

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In this paper, we first investigate the relationship between W-(co)resolutions and X -(co)resolutions for two full subcategories W and X of an abelian category with W ⊆ X . Then some applications are given. In particular, we obtain the stability of the category of C-Gorenstein flat modules under the procedure used to define these entities, which is different from that established by Sather-Wagstaff, Sharif and White. 2010 AMS Mathematics subject classification. Primary 16E05, 16E30, 18G10. Keywords and phrases. X -(co)resolution, (co)generator, semidualizing module, Gorenstein flat module, stability of category.

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Stability of Gorenstein Flat Modules

Cited by 15 publications

References 8 publications

Stability of Gorenstein Flat Categories

Stability of Gorenstein Flat Categories

Gorenstein projective complexes with respect to cotorsion pairs

Resolutions and stability of $C$-Gorenstein flat modules

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