• 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.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.735-744
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• 2024

In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called w-operation, we show that an integral domain R is Krull if and only if for any nonzero proper w-ideal I, the Gorenstein global dimension of the w-factor ring (R/I)_w is zero. Further, we obtain that an integral domain R is Dedekind if and only if for any nonzero proper ideal I, the Gorenstein global dimension of the factor ring R/I is zero.

• 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.

• Journal of the Korean Mathematical Society
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• v.54 no.2
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• pp.427-440
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• 2017

In this paper, in terms of the notions of strongly copure projective modules and the copure projective dimension cpD(R) of a ring R were defined in [12], we show that a domain R has $cpD(R){\leq}1$ if and only if R is a Gorenstein Dedekind domain.

• 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, U_A 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.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1203-1227
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• 2022

In this paper we develop the homological properties of the Gorenstein (𝓛, 𝓐)-flat R-modules 𝓖𝓕_{(𝓕(R),𝓐)} proposed by Gillespie, where the class 𝓐 ⊆ Mod(R^op) sometimes corresponds to a duality pair (𝓛, 𝓐). We study the weak global and finitistic dimensions that come with the class 𝓖𝓕_{(𝓕(R),𝓐)} and show that over a (𝓛, 𝓐)-Gorenstein ring, the functor - ⊗_R - is left balanced over Mod(R^op) × Mod(R) by the classes 𝓖𝓕_{(𝓕(R^op),𝓐)} × 𝓖𝓕_{(𝓕(R),𝓐)}. When the duality pair is (𝓕(R), 𝓕𝓟Inj(R^op)) we recover the G. Yang's result over a Ding-Chen ring, and we see that is new for (Lev(R), AC(R^op)) among others.

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.733-756
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• 2022

In this paper, we introduce the notion of Gorenstein quasiresolving subcategories (denoted by 𝒢𝒬𝓡_𝒳 (𝓐)) in term of quasi-resolving subcategory 𝒳. We define a resolution dimension relative to the Gorenstein quasi-resolving categories 𝒢𝒬𝓡_𝒳 (𝓐). In addition, we study the stability of 𝒢𝒬𝓡_𝒳 (𝓐) and apply the obtained properties to special subcategories and in particular to modules categories. Finally, we use the restricted flat dimension of right B-module M to characterize the finitistic dimension of the endomorphism algebra B of a 𝒢𝒬_𝒳-projective A-module M.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.497-505
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• 2021

Let R → S be a ring homomorphism. The relations of Gorenstein projective dimension with respect to a semidualizing module of homologically bounded complexes between U ⊗LR X and X are considered, where X is an R-complex and U is an S-complex. Some sufficient conditions are given under which the equality ${\mathcal{GP}}_{\tilde{C}}-pd_S(S{\otimes}{L \atop R}X)={\mathcal{GP}}_C-pd_R(X)$ holds. As an application it is shown that the Auslander-Buchsbaum formula holds for GC-projective dimension.

• Journal of the Korean Mathematical Society
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• v.49 no.6
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• pp.1197-1214
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• 2012

The classes of $G_C$ homological modules over commutative ring, where C is a semidualizing module, extend Holm and J${\varnothing}$gensen's notions of C-Gorenstein homological modules to the non-Noetherian setting and generalize the classical classes of homological modules and the classes of Gorenstein homological modules within this setting. On the other hand, transfer of homological properties along ring homomorphisms is already a classical field of study. Motivated by the ideas mentioned above, in this article we will investigate the transfer properties of C and $G_C$ homological dimension.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.779-791
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• 2016

Transfer of homological properties under base change is a classical field of study. Let $R{\rightarrow}S$ be a ring homomorphism. The relations of Gorenstein projective (or Gorenstein injective) dimensions of unbounded complexes between $U{\otimes}^L_RX$(or $RHom_R(X,U)$) and X are considered, where X is an R-complex and U is an S-complex. In addition, some sufficient conditions are given under which the equalities $G-dim_S(U{\otimes}^L_RX)=G-dim_RX+pd_SU$ and $Gid_S(RHom_R(X,U))=G-dim_RX+id_SU$ hold.