• Title/Summary/Keyword: Gorenstein module

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GORENSTEIN MODULES UNDER FROBENIUS EXTENSIONS

  • Kong, Fangdi;Wu, Dejun
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.6
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    • pp.1567-1579
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    • 2020
  • Let R ⊂ S be a Frobenius extension of rings and M a left S-module and let 𝓧 be a class of left R-modules and 𝒚 a class of left S-modules. Under some conditions it is proven that M is a 𝒚-Gorenstein left S-module if and only if M is an 𝓧-Gorenstein left R-module if and only if S ⊗R M and HomR(S, M) are 𝒚-Gorenstein left S-modules. This statement extends a known corresponding result. In addition, the situations of Ding modules, Gorenstein AC modules and projectively coresolved Gorenstein flat modules are considered under Frobenius extensions.

GORENSTEIN FLAT-COTORSION MODULES OVER FORMAL TRIANGULAR MATRIX RINGS

  • Wu, Dejun
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1483-1494
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    • 2021
  • Let A and B be rings and U be a (B, A)-bimodule. If BU has finite flat dimension, UA has finite flat dimension and U ⊗A C is a cotorsion left B-module for any cotorsion left A-module C, then the Gorenstein flat-cotorsion modules over the formal triangular matrix ring $T=\(\array{A&0\\U&B}\)$ are explicitly described. As an application, it is proven that each Gorenstein flat-cotorsion left T-module is flat-cotorsion if and only if every Gorenstein flat-cotorsion left A-module and B-module is flat-cotorsion. In addition, Gorenstein flat-cotorsion dimensions over the formal triangular matrix ring T are studied.

ON OVERRINGS OF GORENSTEIN DEDEKIND DOMAINS

  • Hu, Kui;Wang, Fanggui;Xu, Longyu;Zhao, Songquan
    • Journal of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.991-1008
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    • 2013
  • In this paper, we mainly discuss Gorenstein Dedekind do-mains (G-Dedekind domains for short) and their overrings. Let R be a one-dimensional Noetherian domain with quotient field K and integral closure T. Then it is proved that R is a G-Dedekind domain if and only if for any prime ideal P of R which contains ($R\;:_K\;T$), P is Gorenstein projective. We also give not only an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domain: a 2-DVR. As an application, we prove that a Noetherian domain R is a Warfield domain if and only if for any maximal ideal M of R, $R_M$ is a 2-DVR.

ON GI-FLAT MODULES AND DIMENSIONS

  • Gao, Zenghui
    • Journal of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.203-218
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    • 2013
  • Let R be a ring. A right R-module M is called GI-flat if $Tor^R_1(M,G)=0$ for every Gorenstein injective left R-module G. It is shown that GI-flat modules lie strictly between flat modules and copure flat modules. Suppose R is an $n$-FC ring, we prove that a finitely presented right R-module M is GI-flat if and only if M is a cokernel of a Gorenstein flat preenvelope K ${\rightarrow}$ F of a right R-module K with F flat. Then we study GI-flat dimensions of modules and rings. Various results in [6] are developed, some new characterizations of von Neumann regular rings are given.

THE BONGARTZ'S THEOREM OF GORENSTEIN COSILTING COMPLEXES

  • Hailou Yao ;Qianqian Yuan
    • Journal of the Korean Mathematical Society
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    • v.60 no.6
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    • pp.1337-1364
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    • 2023
  • We describe the Gorenstein derived categories of Gorenstein rings via the homotopy categories of Gorenstein injective modules. We also introduce the concept of Gorenstein cosilting complexes and study its basic properties. This concept is generalized by cosilting complexes in relative homological methods. Furthermore, we investigate the existence of the relative version of the Bongartz's theorem and construct a Bongartz's complement for a Gorenstein precosilting complex.

ON GORENSTEIN COTORSION DIMENSION OVER GF-CLOSED RINGS

  • Gao, Zenghui
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.173-187
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    • 2014
  • In this article, we introduce and study the Gorenstein cotorsion dimension of modules and rings. It is shown that this dimension has nice properties when the ring in question is left GF-closed. The relations between the Gorenstein cotorsion dimension and other homological dimensions are discussed. Finally, we give some new characterizations of weak Gorenstein global dimension of coherent rings in terms of Gorenstein cotorsion modules.

DING PROJECTIVE DIMENSION OF GORENSTEIN FLAT MODULES

  • Wang, Junpeng
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1935-1950
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    • 2017
  • Let R be a Ding-Chen ring. Yang [24] and Zhang [25] asked whether or not every R-module has finite Ding projective or Ding injective dimension. In this paper, we give a new characterization of that all modules have finite Ding projective and Ding injective dimension in terms of the relationship between Ding projective and Gorenstein flat modules. We also give an example to obtain negative answer to the above question.

(𝒱, 𝒲, 𝑦, 𝒳)-GORENSTEIN COMPLEXES

  • Yanjie Li;Renyu Zhao
    • Journal of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.603-620
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    • 2024
  • Let 𝒱, 𝒲, 𝑦, 𝒳 be four classes of left R-modules. The notion of (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein R-complexes is introduced, and it is shown that under certain mild technical assumptions on 𝒱, 𝒲, 𝑦, 𝒳, an R-complex 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein if and only if the module in each degree of 𝑴 is (𝒱, 𝒲, 𝑦, 𝒳)-Gorenstein and the total Hom complexs HomR(𝒀, 𝑴), HomR(𝑴, 𝑿) are exact for any ${\mathbf{Y}}\,{\in}\,{\tilde{\mathcal{Y}}}$ and any ${\mathbf{X}}\,{\in}\,{\tilde{\mathcal{X}}}$. Many known results are recovered, and some new cases are also naturally generated.

GORENSTEIN-INJECTORS, GORENSTEIN-FLATORS

  • Gu, Qinqin;Zhu, Xiaosheng;Zhou, Wenping
    • Journal of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.691-704
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    • 2010
  • Over a ring R, let $P_R$ be a finitely generated projective right R-module. Then we define the G-injector (G-projector) if $P_R$ preservers Gorenstein injective modules (Gorenstein projective modules), the Gflator if $P_R$ preservers Gorenstein flat modules. G-injector (G-flator) and G-injector are characterized focus primarily on the cases where R is a Gorenstein ring, and under this condition we also study the relations between the injector (projector, flator) and the G-injector (G-projector, G-flator). Over any ring we also give the characteristics of G-injector (G-flator) by the Gorenstein injective (Gorenstein flat) dimensions of modules.