• Korean Journal of Mathematics
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• v.22 no.1
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• pp.181-206
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• 2014

Let $R={\bigoplus}_{\alpha{\in}\Gamma}R_{\alpha}$ be a graded integral domain graded by an arbitrary grading torsionless monoid ${\Gamma}$, and ⋆ be a semistar operation on R. In this paper we define and study the graded integral domain analogue of ⋆-Nagata and Kronecker function rings of R with respect to ⋆. We say that R is a graded Pr$\ddot{u}$fer ⋆-multiplication domain if each nonzero finitely generated homogeneous ideal of R is ⋆$_f$-invertible. Using ⋆-Nagata and Kronecker function rings, we give several different equivalent conditions for R to be a graded Pr$\ddot{u}$fer ⋆-multiplication domain. In particular we give new characterizations for a graded integral domain, to be a $P{\upsilon}MD$.

• 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).

• Korean Journal of Mathematics
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• v.22 no.4
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• pp.591-598
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• 2014

Let D be an integral domain with quotient field K. In [1], the authors called D a finitely valuative domain if, for each $0{\neq}u{\in}K$, there is a saturated chain of rings $D=D_0{\varsubsetneq}D_1{\varsubsetneq}{\cdots}{\subseteq}$ $D_n=D[x]$, where x = u or $u^{-1}$. They then studied some properties of finitely valuative domains. For example, they showed that the integral closure of a finitely valuative domain is a Pr$\ddot{u}$fer domain. In this paper, we introduce the notion of finitely t-valuative domains, which is the t-operation analog of finitely valuative domains, and we then generalize some properties of finitely valuative domains.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.393-410
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• 2013

Let $R{\subseteq}T$ be an extension of integral domains, X be an indeterminate over T, and R[X] and T[X] be polynomial rings. Then $R{\subseteq}T$ is said to be LCM-stable if $(aR{\cap}bR)T=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$. Let $w_A$ be the so-called $w$-operation on an integral domain A. In this paper, we introduce the notions of $w(e)$- and $w$-LCM-stable extensions: (i) $R{\subseteq}T$ is $w(e)$-LCM-stable if $((aR{\cap}bR)T)_{w_T}=aT{\cap}bT$ for all $0{\neq}a,b{\in}R$ and (ii) $R{\subseteq}T$ is $w$-LCM-stable if $((aR{\cap}bR)T)_{w_R}=(aT{\cap}bT)_{w_R}$ for all $0{\neq}a,b{\in}R$. We prove that LCM-stable extensions are both $w(e)$-LCM-stable and $w$-LCM-stable. We also generalize some results on LCM-stable extensions. Among other things, we show that if R is a Krull domain (resp., $P{\upsilon}MD$), then $R{\subseteq}T$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable) if and only if $R[X]{\subseteq}T[X]$ is $w(e)$-LCM-stable (resp., $w$-LCM-stable).