• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1689-1701
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• 2019

Let ${\star}$ be a semistar operation on the integral domain D. In this paper, we prove that D is a $G-{\tilde{\star}}-GCD$ domain if and only if D[X] is a $G-{\star}_1-GCD$ domain if and only if the Nagata ring of D with respect to the semistar operation ${\tilde{\star}}$, $Na(D,{\star}_f)$ is a G-GCD domain if and only if $Na(D,{\star}_f)$ is a GCD domain, where ${\star}_1$ is the semistar operation on D[X] introduced by G. Picozza [12].

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1179-1192
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• 2009

In this note we give a new generalization of the notions of $Pr{\ddot{U}}fer$ domain and PvMD which uses quasi semistar invertibility, the "quasi P$\star$MD", and compare them with the P$\star$MD. We show in particular that the problem of when a quasi P$\star$MD is a P$\star$MD is strictly related to the problem of the descent to subrings of the P$\star$MD property and we give necessary and sufficient conditions.

• Korean Journal of Mathematics
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• v.22 no.3
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• pp.455-462
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• 2014

Let D be an integral domain with Spec(D) finite, K the quotient field of D, [D,K] the set of rings between D and K, and SFc(D) the set of semistar operations of finite character on D. It is well known that |Spec(D)| ${\leq}$ |SFc(D)|. In this paper, we prove that |Spec(D)| = |SFc(D)| if and only if D is a valuation domain, if and only if |Spec(D)| = |[D,K]|. We also study integral domains D such that |Spec(D)|+1 = |SFc(D)|.

• 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 ${\mid}SStar(R){\mid}=14$.

• 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 $P{\upsilon}MD$.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.397-408
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• 2018

It is well-known that quasi-$Pr{\ddot{u}}fer$ domains are characterized as those domains D, such that every extension of D inside its quotient field is a primitive extension and that primitive extensions are characterized in terms of INC-extensions. Let $R={\bigoplus}_{{\alpha}{{\in}}{\Gamma}}$ $R_{\alpha}$ be a graded integral domain graded by an arbitrary torsionless grading monoid ${\Gamma}$ and ${\star}$ be a semistar operation on R. The main purpose of this paper is to give new characterizations of gr-${\star}$-quasi-$Pr{\ddot{u}}fer$ domains in terms of graded primitive and INC-extensions. Applications include new characterizations of UMt-domains.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.187-204
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• 2018

Let $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}\;R_{\alpha}$ be a graded integral domain, H be the set of nonzero homogeneous elements of R, and ${\star}$ be a semistar operation on R. The purpose of this paper is to study the properties of $quasi-Pr{\ddot{u}}fer$ and UMt-domains of graded integral domains. For this reason we study the graded analogue of ${\star}-quasi-Pr{\ddot{u}}fer$ domains called $gr-{\star}-quasi-Pr{\ddot{u}}fer$ domains. We study several ring-theoretic properties of $gr-{\star}-quasi-Pr{\ddot{u}}fer$ domains. As an application we give new characterizations of UMt-domains. In particular it is shown that R is a $gr-t-quasi-Pr{\ddot{u}}fer$ domain if and only if R is a UMt-domain if and only if RP is a $quasi-Pr{\ddot{u}}fer$ domain for each homogeneous maximal t-ideal P of R. We also show that R is a UMt-domain if and only if H is a t-splitting set in R[X] if and only if each prime t-ideal Q in R[X] such that $Q{\cap}H ={\emptyset}$ is a maximal t-ideal.