• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.209-219
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• 2005

Tor-torsion theory was defined by Jan Trlifaj in 2000. In this paper we introduce the notion of Co envelopes, CoCovers and Tor-generators as dual of envelopes, covers and generators in cotorsion(Ext-torsion) theory and deduce that each R-module has a projective and a cotorsion coprecover.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.531-540
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• 2015

Let (R, m) be a Noetherian local ring, I an ideal of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{>k}(I,A)$ the supremum of length of A-cosequence in dimension > k in I defined by Nhan-Hoang [8]. It is shown that for all $t{\leq}r$ the sets $$(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/I^n,A)))_{{\geq}k}\;and\\(\bigcup_{i=0}^{t}Att_R(Tor_i^R(R/(a_1^{n_1},{\cdots},a_l^{n_l}),A)))_{{\geq}k}$$ are independent of the choice of $n,n_1,{\cdots},n_l$ for any system of generators ($a_1,{\cdots},a_l$) of I.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.763-780
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• 2020

Let R be a commutative ring. An R-module M is said to be w-flat if Tor ^R₁ (M, N) is GV -torsion for any R-module N. It is known that every flat module is w-flat, but the converse is not true in general. The w-flat dimension of a module is defined in terms of w-flat resolutions. In this paper, we study the w-flat dimension of an injective w-module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) flat modules and the w-copure flat dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We also study change of rings theorems for the w-copure flat dimension in various contexts. Finally some illustrative examples regarding the introduced concepts are given.

• 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.50 no.1
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• pp.11-24
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• 2013

Let m and n be fixed positive integers and M a right R-module. Recall that M is said to be ($m$, $n$)-injective if $Ext^1$(P, M) = 0 for any ($m$, $n$)-presented right R-module P; M is said to be ($m$, $n$)-flat if $Tor_1$(N, P) = 0 for any ($m$, $n$)-presented left R-module P. In terms of some derived functors, relative injective or relative flat resolutions and dimensions are investigated. As applications, some new characterizations of von Neumann regular rings and p.p. rings are given.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1221-1233
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• 2018

Let R be a commutative ring with non-zero identity and let NN(R) = {I | I is a nonnil ideal of R}. Let M be an R-module and let ${\phi}-tor(M)=\{x{\in}M{\mid}Ix=0\text{ for some }I{\in}NN(R)\}$. If ${\phi}or(M)=M$, then M is called a ${\phi}$-torsion module. An R-module M is said to be ${\phi}$-flat, if $0{\rightarrow}{A{\otimes}_R}\;{M{\rightarrow}B{\otimes}_R}\;{M{\rightarrow}C{\otimes}_R}\;M{\rightarrow}0$ is an exact R-sequence, for any exact sequence of R-modules $0{\rightarrow}A{\rightarrow}B{\rightarrow}C{\rightarrow}0$, where C is ${\phi}$-torsion. In this paper, the concepts of NRD-submodules and NP-submodules are introduced, and the ${\phi}$-flat modules over a ${\phi}-Pr{\ddot{u}}fer$ ring are investigated.

• Kyungpook Mathematical Journal
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• v.63 no.3
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• pp.333-344
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• 2023

In this paper, we generalize the notions uniformly S-flat, briefly u-S-flat, modules and dimensions. We introduce and study the notions of weak u-S-flat modules. An R-module M is said to be weak u-S-flat if Tor^R₁ (R/I, M) is u-S-torsion for any ideal I of R. This new class of modules will be used to characterize u-S-von Neumann regular rings. Hence, we introduce the weak u-S-flat dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed.

• Communications of the Korean Mathematical Society
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• v.25 no.4
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• pp.497-510
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• 2010

Let R be a ring and n a fixed non-negative integer. $\cal{TI}_n$ (resp. $\cal{TF}_n$) denotes the class of all right R-modules of FGT-injective dimensions at most n (resp. all left R-modules of FGT-flat dimensions at most n). We prove that, if R is a right $\prod$-coherent ring, then every right R-module has a $\cal{TI}_n$-cover and every left R-module has a $\cal{TF}_n$-preenvelope. A right R-module M is called n-TI-injective in case $Ext^1$(N,M) = 0 for any $N\;{\in}\;\cal{TI}_n$. A left R-module F is said to be n-TI-flat if $Tor_1$(N, F) = 0 for any $N\;{\in}\;\cal{TI}_n$. Some properties of n-TI-injective and n-TI-flat modules and their relations with $\cal{TI}_n$-(pre)covers and $\cal{TF}_n$-preenvelopes are also studied.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.273-280
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• 2024

Let I be an ideal of a commutative Noetherian semi-local ring R and M be an R-module. It is shown that if dim M ≤ 2 and Supp_R M ⊆ V (I), then M is I-weakly cofinite if (and only if) the R-modules Hom_R(R/I, M) and Ext¹_R(R/I, M) are weakly Laskerian. As a consequence of this result, it is shown that the category of all I-weakly cofinite modules X with dim X ≤ 2, forms an Abelian subcategory of the category of all R-modules. Finally, it is shown that if dim R/I ≤ 2, then for each pair of finitely generated R-modules M and N and each pair of the integers i, j ≥ 0, the R-modules Tor^R_i(N, H^j_I(M)) and Extⁱ_R(N, H^j_I(M)) are I-weakly cofinite.

• Journal of the Korean Society for Precision Engineering
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• v.14 no.2
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• pp.48-57
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• 1997

A computer aided analysis and design system is developed for analyzing the driveline torsional vibration of a vehicle. Torsional vibration characteristics of driveline component are investigated and 10 types of module are developed. They can be connected together to represent any driveline configuration. During assembly process simulation program is generated. It is implemented using C++language. A driveline tor- sional vibration system at full load driving condition and idle rattle system are modeled and simulated with this system. Their responses for engine torque excitation are evaluated on time and frequency domain, and the results are compared with test results favorably. This system makes it simpler and easier for design and analysis engineer to model and analyse the driveline system.