• Journal of the Korean Mathematical Society
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• v.38 no.5
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• pp.915-935
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• 2001

Let V be any valuation domain and let E be a subset of the quotient field K of V. We study the ring of integer-valued polynomials on E, that is, Int(E, V)={f$\in$K[X]｜f(E)⊆V}. We show that, if E is precompact, then Int(E, V) has many properties similar to those of the classical ring Int(Z).In particular, Int(E, V) is dense in the ring of continuous functions C(E, V); each finitely generated ideal of Int(E, V) may be generated by two elements; and finally, Int(E, V) is a Prufer domain.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.971-983
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• 2023

In this paper, we prove that a domain R is an FGV-domain if every finitely generated torsion-free R-module is strongly copure projective, and a coherent domain is an FGV-domain if and only if every finitely generated torsion-free R-module is strongly copure projective. To do this, we characterize G-Prüfer domains by G-flat modules, and we prove that a domain is G-Prüfer if and only if every submodule of a projective module is G-flat. Also, we study the D + M construction of G-Prüfer domains. It is seen that there exists a non-integrally closed G-Prüfer domain that is neither Noetherian nor divisorial.

• Bulletin of the Korean Mathematical Society
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• v.46 no.5
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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).

• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.259-268
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• 1998

We give several characterizations of a TV-PVMD and we show that the localization R[X;S]$_{N_{\upsilon}}$ of a semigroup ring R[X;S] is a TV-PVMD if and only if R is a TV-PVMD where $N_{\upsilon}\;=\;\{f\;{\in}\;R[X]{\mid}(A_f)_{\upsilon} = R\}$ and S is a torsion free cancellative semigroup with zero.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1485-1495
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• 2019

The purpose of this paper is to compute the number of semistar operations of certain classes of finite dimensional $Pr{\ddot{u}}fer$ domains. We prove that ${\mid}SStar(R){\mid}={\mid}Star(R){\mid}+{\mid}Spec(R){\mid}+ {\mid}Idem(R){\mid}$ where Idem(R) is the set of all nonzero idempotent prime ideals of R if and only if R is a $Pr{\ddot{u}}fer$ domain with Y -graph spectrum, that is, R is a $Pr{\ddot{u}}fer$ domain with exactly two maximal ideals M and N and $Spec(R)=\{(0){\varsubsetneq}P_1{\varsubsetneq}{\cdots}{\varsubsetneq}P_{n-1}{\varsubsetneq}M,N{\mid}P_{n-1}{\varsubsetneq}N\}$. We also characterize non-local $Pr{\ddot{u}}fer$ domains R such that ${\mid}SStar(R){\mid}=7$, respectively ${\mid}SStar(R){\mid}=14$.