• 대한수학회논문집
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• 제33권4호
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• pp.1083-1096
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• 2018

Let R be an associative ring. We define a subset $S_R$ of R as $S_R=\{a{\in}R{\mid}aRa=(0)\}$ and call it the source of semiprimeness of R. We first examine some basic properties of the subset $S_R$ in any ring R, and then define the notions such as R being a ${\mid}S_R{\mid}$-reduced ring, a ${\mid}S_R{\mid}$-domain and a ${\mid}S_R{\mid}$-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite ${\mid}S_R{\mid}$-domain is necessarily unitary, and is in fact a ${\mid}S_R{\mid}$-division ring. However, we provide an example showing that a finite ${\mid}S_R{\mid}$-division ring does not need to be commutative. All possible values for characteristics of unitary ${\mid}S_R{\mid}$-reduced rings and ${\mid}S_R{\mid}$-domains are also determined.

• 대한수학회보
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• 제52권4호
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• pp.1285-1295
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• 2015

An SM domain is an integral domain which satisfies the ascending chain condition on w-ideals. Then an SM domain also satisfies the descending chain condition on those chains of v-ideals whose intersection is not zero. In this paper, a study is begun to extend these properties to commutative rings with zero divisors. A $Q_0$-SM ring is defined to be a ring which satisfies the ascending chain condition on semiregular w-ideals and satisfies the descending chain condition on those chains of semiregular v-ideals whose intersection is semiregular. In this paper, some properties of $Q_0$-SM rings are discussed and examples are provided to show the difference between $Q_0$-SM rings and SM rings and the difference between $Q_0$-SM rings and $Q_0$-Mori rings.

• Korean Journal of Mathematics
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• 제19권3호
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• pp.255-261
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• 2011

Let D be an integral domain, S be a torsion-free grading monoid such that the quotient group of S is of type (0, 0, 0, ${\ldots}$), and D[S] be the semigroup ring of S over D. We show that D[S] is an H-domain if and only if D is an H-domain and each maximal t-ideal of S is a $v$-ideal. We also show that if $\mathbb{R}$ is the eld of real numbers and if ${\Gamma}$ is the additive group of rational numbers, then $\mathbb{R}[{\Gamma}]$ is not an H-domain.

• 대한수학회지
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• 제48권1호
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• pp.191-205
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• 2011

It is well known that for a Krull domain R, the divisor class group of R is a torsion group if and only if every subintersection of R is a ring of quotients. Thus a natural question is that under what conditions, for a non-Krull domain R, every (t-)subintersection (resp., t-linked overring) of R is a ring of quotients or every (t-)subintersection (resp., t-linked overring) of R is at. To address this question, we introduce the notions of *-compact packedness and *-coprime packedness of (an ideal of) an integral domain R for a star operation * of finite character, mainly t or w. We also investigate the t-theoretic analogues of related results in the literature.

• Kyungpook Mathematical Journal
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• 제57권3호
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• pp.433-439
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• 2017

It is well known that an integral domain D is a Dedekind domain if and only if D is a Noetherian almost Dedekind domain. In this paper, we show that an integral domain D is a Dedekind domain if and only if D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology. We also give a new characterization of ZPI-rings.

• Kyungpook Mathematical Journal
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• 제63권4호
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• pp.521-537
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• 2023

A module M is called a simple-direct-injective module if, whenever A and B are simple submodules of M with A ≅ B and B is a direct summand of M, then A is a direct summand of M. Some new characterizations of these modules are proved. The structure of simple-direct-injective modules over a commutative Dedekind domain is fully determined. Also, some relevant counterexamples are indicated to show that a left simple-direct-injective ring need not be right simple-direct-injective.

• 대한수학회보
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• 제46권5호
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• pp.1013-1018
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• 2009

Let * be an e.a.b. star operation on an integrally closed domain D, and let $K\gamma$(D, *) be the Kronecker function ring of D. We show that if D is a P*MD, then the mapping $D_{\alpha}{\mapsto}K{\gamma}(D_{\alpha},\;{\upsilon})$ is a bijection from the set {$D_{\alpha}$} of *-linked overrings of D into the set of overrings of $K{\gamma}(D,\;{\upsilon})$. This is a generalization of [5, Proposition 32.19] that if D is a Pr$\ddot{u}$fer domain, then the mapping $D_{\alpha}{\mapsto}K_{\gamma}(D_{\alpha},\;b)$ is a one-to-one mapping from the set {$D_{\alpha}$} of overrings of D onto the set of overrings of $K_{\gamma}$(D, b).

• Korean Journal of Mathematics
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• 제28권1호
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• pp.65-73
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• 2020

This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

• Korean Journal of Mathematics
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• 제18권1호
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• pp.37-43
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• 2010

Let D be an integral domain, X be an indeterminate over D, and $N_v=\{f{\in}D[X]{\mid}(A_f)_v=D\}$. In this paper, we introduce the concept of t-locally divided domains, and we then prove that $D[X]_{N_v}$ is a locally divided domain if and only if D is a t-locally divided UMT-domain, if and only if D[X] is a t-locally divided domain.

• East Asian mathematical journal
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• 제36권3호
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• pp.417-424
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• 2020

Huang et al. proved that the n by n upper triangular matrix ring over a domain is weakly reversible-over-center by using the property of regular matrices. In this article we provide a concrete proof which is able to be available in the related study of centers. Next we extend an example of weakly reversible-over-center, which was argued by Huang et al., to the general case.