• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.895-903
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• 2023

In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring R is 2-strongly Gorenstein hereditary if and only if every ideal of R is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.

• 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.58 no.4
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• pp.939-958
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• 2021

Some properties of strongly Gorenstein C-projective, C-injective and C-flat modules are studied, mainly considering these properties under change of rings. Specifically, the completions of rings, the localizations and the polynomial rings are considered.

• 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.

• 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.49 no.1
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• pp.31-47
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• 2012

The so-called Ding-Chen ring is an n-FC ring which is both left and right coherent, and has both left and right self FP-injective dimensions at most n for some non-negative integer n. In this paper, we investigate the classes of the so-called Ding projective, Ding injective and Gorenstein at modules and show that some homological properties of modules over Gorenstein rings can be generalized to the modules over Ding-Chen rings. We first consider Gorenstein at and Ding injective dimensions of modules together with Ding injective precovers. We then discuss balance of functors Hom and tensor.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.871-883
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• 2006

We show that the Auslander index is the same as the column invariant over Gorenstein local rings. We also show that Ding's conjecture ([13]) holds for an isolated non-Gorenstein ring A satisfying a certain condition which seems to be weaker than the condition that the associated graded ring of A is Cohen-Macaulay.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.373-382
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• 2019

In this note, we mainly discuss strongly Gorenstein hereditary rings. We prove that for any ring, the class of SG-projective modules and the class of G-projective modules coincide if and only if the class of SG-projective modules is closed under extension. From this we get that a ring is an SG-hereditary ring if and only if every ideal is G-projective and the class of SG-projective modules is closed under extension. We also give some examples of domains whose ideals are SG-projective.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.989-996
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• 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper we study the amalgamated duplication ring $R{\bowtie}I$ which is introduced by D'Anna and Fontana. It is shown that if R is generically Cohen-Macaulay (resp. generically Gorenstein) and I is generically maximal Cohen-Macaulay (resp. generically canonical module), then $R{\bowtie}I$ is generically Cohen-Macaulay (resp. generically Gorenstein). We also de ned generically quasi-Gorenstein ring and we investigate when $R{\bowtie}I$ is generically quasi-Gorenstein. In addition, it is shown that $R{\bowtie}I$ is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if R is approximately Gorenstein, then $R{\bowtie}I$ is approximately Gorenstein.