• Korean Journal of Mathematics
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• 제7권2호
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• pp.167-169
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• 1999

Let R be a Noetherian domain. It is proved that $t$-dimR = 1 if and only if each (proper if R is not a valuation domain) $t$-linked overring D of R is of $t$-dimD = 1 if and only if each integrally closed $t$-linked overring of R is a Krull domain.

• 대한수학회논문집
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• 제38권1호
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• pp.11-19
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• 2023

Let 𝓗₀ be the set of rings R such that Nil(R) = Z(R) is a divided prime ideal of R. The concept of maximal non φ-chained subrings is a generalization of maximal non valuation subrings from domains to rings in 𝓗₀. This generalization was introduced in [20] where the authors proved that if R ∈ 𝓗₀ is an integrally closed ring with finite Krull dimension, then R is a maximal non φ-chained subring of T(R) if and only if R is not local and |[R, T(R)]| = dim(R) + 3. This motivates us to investigate the other natural numbers n for which R is a maximal non φ-chained subring of some overring S. The existence of such an overring S of R is shown for 3 ≤ n ≤ 6, and no such overring exists for n = 7.

• 대한수학회보
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• 제38권3호
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• pp.543-549
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• 2001

We show that R[X] is a Krull (Resp. factorial) ring if and only if R is a normal Krull (resp, factorial) ring with a finite number of minimal prime ideals if and only if R is a Krull (resp. factorial) ring with a finite number of minimal prime ideals and R(sub)M is an integral domain for every maximal ideal M of R. As a corollary, we have that if R[X] is a Krull (resp. factorial) ring and if D is a Krull (resp. factorial) overring of R, then D[X] is a Krull (resp. factorial) ring.

• 대한수학회보
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• 제49권1호
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• pp.205-211
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• 2012

Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dim $R\leq2$, then A is a graded Noetherian domain with h-dim $A\leq2$. This is a generalization of the well-know theorem that a Krull overring of a Noetherian domain with dimension $\leq2$ is also a Noetherian domain with dimension $\leq2$.

• 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.

• 충청수학회지
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• 제24권3호
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• pp.601-607
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• 2011

Let D be an integral domain, X be an indeterminate over D, and D[X] be the polynomial ring over D. We show that D is an almost weakly factorial PvMD if and only if D + XDS[X] is an integrally closed almost GCD-domain for each (saturated) multiplicative subset S of D, if and only if $D+XD_1[X]$ is an integrally closed almost GCD-domain for any t-linked overring $D_1$ of D, if and only if $D_1+XD_2[X]$ is an integrally closed almost GCD-domain for all t-linked overrings $D_1{\subseteq}D_2$ of D.

• 대한수학회지
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• 제59권3호
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• pp.571-594
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• 2022

Let D be an integral domain with quotient field K, Pic(D) be the ideal class group of D, and X be an indeterminate. A polynomial overring of D means a subring of K[X] containing D[X]. In this paper, we study almost Dedekind domains which are polynomial overrings of a principal ideal domain D, defined by the intersection of K[X] and rank-one discrete valuation rings with quotient field K(X), and their ideal class groups. Next, let ℤ be the ring of integers, ℚ be the field of rational numbers, and 𝔊f be the set of finitely generated abelian groups (up to isomorphism). As an application, among other things, we show that there exists an overring R of ℤ[X] such that (i) R is a Bezout domain, (ii) R∩ℚ[X] is an almost Dedekind domain, (iii) Pic(R∩ℚ[X]) = $\oplus_{G{\in}G_{f}}$ G, (iv) for each G ∈ 𝔊f, there is a multiplicative subset S of ℤ such that RS ∩ ℚ[X] is a Dedekind domain with Pic(RS ∩ ℚ[X]) = G, and (v) every invertible integral ideal I of R ∩ ℚ[X] can be written uniquely as I = XnQe11···Qekk for some integer n ≥ 0, maximal ideals Qi of R∩ℚ[X], and integers ei ≠ 0. We also completely characterize the almost Dedekind polynomial overrings of ℤ containing Int(ℤ).

• 대한수학회지
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• 제45권5호
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• pp.1405-1416
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• 2008

Let D be an integral domain, X an indeterminate over D, $N_v = \{f{\in}D[X]|(A_f)_v=D\}.$. Among other things, we introduce the concept of t-locally PVDs and prove that $D[X]N_v$ is a locally PVD if and only if D is a t-locally PVD and a UMT-domain, if and only if D[X] is a t-locally PVD, if and only if each overring of $D[X]N_v$ is a locally PVD.

• 대한수학회보
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• 제57권6호
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• pp.1351-1365
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• 2020

Let S and T be w-linked extension domains of a domain R with S ⊆ T. In this paper, we define what satisfying the w_R-GD property for S ⊆ T means and what being w_R- or w-GD domains for T means. Then some sufficient conditions are given for the w_R-GD property and w_R-GD domains. For example, if T is w_R-integral over S and S is integrally closed, then the w_R-GD property holds. It is also given that S is a w_R-GD domain if and only if S ⊆ T satisfies the w_R-GD property for each w_R-linked valuation overring T of S, if and only if S ⊆ (S[u])_w satisfies the w_R-GD property for each element u in the quotient field of S, if and only if S_𝔪 is a GD domain for each maximal w_R-ideal 𝔪 of S. Then we focus on discussing the relationship among GD domains, w-GD domains, w_R-GD domains, Prüfer domains, PνMDs and Pw_RMDs, and also provide some relevant counterexamples. As an application, we give a new characterization of Pw_RMDs. We show that S is a Pw_RMD if and only if S is a w_R-GD domain and every w_R-linked overring of S that satisfies the w_R-GD property is w_R-flat over S. Furthermore, examples are provided to show these two conditions are necessary for Pw_RMDs.

• 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).