• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.65-76
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• 2006

Let $\tau$ be a hereditary torsion theory and let p be a prime ideal of a commutative ring R. We study the existence of (minimal) $\tau-injective$ submodules of the injective hull of R/p.

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.821-841
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• 2022

In this paper, we introduce and study regular rings relative to the hereditary torsion theory w (a special case of a well-centered torsion theory over a commutative ring), called w-regular rings. We focus mainly on the w-regularity for w-coherent rings and w-Noetherian rings. In particular, it is shown that the w-coherent w-regular domains are exactly the Prüfer v-multiplication domains and that an integral domain is w-Noetherian and w-regular if and only if it is a Krull domain. We also prove the w-analogue of the global version of the Serre-Auslander-Buchsbaum Theorem. Among other things, we show that every w-Noetherian w-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak w-projective dimension of a w-Noetherian ring is 0, 1, or ∞.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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• pp.867-871
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• 2009

A module M over a ring R is said to satisfy (P) if every generating set of M contains an independent generating set. The following results are proved; (1) Let $\tau$ = ($\mathbb{T}_\tau,\;\mathbb{F}_\tau$) be a hereditary torsion theory such that $\mathbb{T}_\tau$ $\neq$ Mod-R. Then every $\tau$-torsionfree R-module satisfies (P) if and only if S = R/$\tau$(R) is a division ring. (2) Let $\mathcal{K}$ be a hereditary pre-torsion class of modules. Then every module in $\mathcal{K}$ satisfies (P) if and only if either $\mathcal{K}$ = {0} or S = R/$Soc_\mathcal{K}$(R) is a division ring, where $Soc_\mathcal{K}$(R) = $\cap${I 4\leq$ $R_R$ : R/I$\in\mathcal{K}$}.

• Journal of the Korean Mathematical Society
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• v.31 no.3
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• pp.439-456
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• 1994

Let $\tau$ be a given hereditary torsion theory for left R-module category R-Mod. The class of all $\tau$-torsion left R-modules, denoted by T is closed under homomorphic images, submodules, direct sums and extensions. And the class of all $\tau$-torsionfree left R-modules, denoted by $F$, is closed under submodules, injective hulls, direct products, and isomorphic copies ([3], Proposition 1.7 and 1.10).

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1617-1642
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• 2019

In this paper, we study a theory for the structure of ${\tau}_w$-Loewy series of modules over commutative rings, where ${\tau}_w$ is the hereditary torsion theory induced by the so-called w-operation, and explore the relationship between ${\tau}_w$-Loewy modules and w-Artinian modules.

• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.937-954
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• 2001

This paper is a natural continuation of [2], [3], [4] and [5]. Localization techniques for modular lattices are developed. These techniques are applied to study liftings of linear order types from quotient lattices and to find Г-dense sets in certain lattices without Г-deviation in the sense of [4], where Г is a set of indecomposable linear order types.