• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.95-103
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• 2019

In this paper we study certain modules whose prime spectrums are Noetherian or/and spectral spaces. In particular, we investigate the relationship between topological properties of prime spectra of torsion modules and algebraic properties of them.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1051-1066
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• 2013

By utilizing known characterizations of ${\omega}$-Noetherian rings in terms of injective modules, we give more characterizations of ${\omega}$-Noetherian rings. More precisely, we show that a commutative ring R is ${\omega}$-Noetherian if and only if the direct limit of GV -torsion-free injective R-modules is injective; if and only if every R-module has a GV -torsion-free injective (pre)cover; if and only if the direct sum of injective envelopes of ${\omega}$-simple R-modules is injective; if and only if the essential extension of the direct sum of GV -torsion-free injective R-modules is the direct sum of GV -torsion-free injective R-modules; if and only if every $\mathfrak{F}_{w,f}(R)$-injective ${\omega}$-module is injective; if and only if every GV-torsion-free R-module admits an $i$-decomposition.

• Honam Mathematical Journal
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• v.28 no.4
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• pp.499-511
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• 2006

It is well-known that every finitely generated torsion-free module over a principal ideal domain is free. This will be generalized. We deal with ideals of the finite, external direct product of certain rings. Finally, if M is a torsion-free, finitely generated module over a reduced, Noetherian ring A, then we prove that Ms is a projective module over As, where $S=A{\setminus}(A)$.

• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.645-652
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• 2006

The purpose of this paper is to describe some characterizations on rational and f-rational extensions of modules, to determine the forms of f-rational extensions of given f-torsion free module for a left exact radical t and investigate the maximal t-rational extensions of modules.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1041-1052
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• 2010

In this paper, strongly cotorsion (torsion-free) modules are studied and strongly cotorsion (torsion-free) dimension is introduced. It is shown that every module has a special $\mathcal{SC}_n$-preenvelope and an ST $\mathcal{F}_n$-cover for any $n\;{\in}\;\mathbb{N}$ based on some results of cotorsion pairs from [9]. Some characterizations of strongly cotorsion (torsion-free) dimension of a module are given.

• 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.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.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.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.577-587
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• 2002

Let A be a commutative ring and M an Artinian .A-module. Let $\sigma$ be a torsion radical functor and (T, F) it's corresponding partition of Spec(A) In [1] the concept of Cohen-Macauly modules was generalized . In this paper we shall define $\sigma$-co-Cohen-Macaulay (abbr. $\sigma$-co-CM). Indeed this is one of the aims of this paper, we obtain some satisfactory properties of such modules. An-other aim of this paper is to generalize the concept of cograde by using the left derived functor $U^{\alpha}$$_{I}$(-) of the $\alpha$-adic completion functor, where a is contained in Jacobson radical of A.A.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.1-14
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• 1998

In this short note we study the torsion theories over a commutative ring R and discuss a relative dimension related to such theories for R-modules. Let ${\sigma}$ be a torsion functor and (T, F) be its corresponding partition of Spec(R). The concept of ${\sigma}$-Cohen Macaulay (abbr. ${\sigma}$-CM) module is defined and some of the main points concerning the usual Cohen-Macaulay modules are extended. In particular it is shown that if M is a non-zero ${\sigma}$-CM module over R and S is a multiplicatively closed subset of R such that, for all minimal element of T, $S{\cap}p={\emptyset}$, then $S^{-1}M$ is a $S^{-1}{\sigma}$-CM module over $S^{-1}$R, where $S^{-1}{\sigma}$ is the direct image of ${\sigma}$ under the natural ring homomorphism $R{\longrightarrow}S^{-1}R$.