• 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.

• Journal of the Korean Mathematical Society
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• v.54 no.6
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• pp.1733-1757
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• 2017

Let $R={\oplus}_{{\alpha}{\in}{\Gamma}}R_{\alpha}$ be an integral domain graded by an arbitrary torsionless grading monoid ${\Gamma}$, ${\bar{R}}$ be the integral closure of R, H be the set of nonzero homogeneous elements of R, C(f) be the fractional ideal of R generated by the homogeneous components of $f{\in}R_H$, and $N(H)=\{f{\in}R{\mid}C(f)_v=R\}$. Let $R_H$ be a UFD. We say that a nonzero prime ideal Q of R is an upper to zero in R if $Q=fR_H{\cap}R$ for some $f{\in}R$ and that R is a graded UMT-domain if each upper to zero in R is a maximal t-ideal. In this paper, we study several ring-theoretic properties of graded UMT-domains. Among other things, we prove that if R has a unit of nonzero degree, then R is a graded UMT-domain if and only if every prime ideal of $R_{N(H)}$ is extended from a homogeneous ideal of R, if and only if ${\bar{R}}_{H{\backslash}Q}$ is a graded-$Pr{\ddot{u}}fer$ domain for all homogeneous maximal t-ideals Q of R, if and only if ${\bar{R}}_{N(H)}$ is a $Pr{\ddot{u}}fer$ domain, if and only if R is a UMT-domain.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.455-462
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• 2013

Let D be an integral domain, S be a saturated multi-plicative subset of D such that $D_S$ is a factorial domain, $\{X_{\alpha}\}$ be a nonempty set of indeterminates, and $D[\{X_{\alpha}\}]$ be the polynomial ring over D. We show that S is a splitting (resp., almost splitting, t-splitting) set in D if and only if every nonzero prime t-ideal of D disjoint from S is principal (resp., contains a primary element, is t-invertible). We use this result to show that $D{\backslash}\{0\}$ is a splitting (resp., almost splitting, t-splitting) set in $D[\{X_{\alpha}\}]$ if and only if D is a GCD-domain (resp., UMT-domain with $Cl(D[\{X_{\alpha}\}]$ torsion UMT-domain).

• Korean Journal of Mathematics
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• v.18 no.1
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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.

• Journal of the Korean Mathematical Society
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• v.45 no.5
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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.

• Korean Journal of Mathematics
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• v.19 no.4
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• pp.343-349
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• 2011

Let D be an integral domain, $D^w$ be the $w$-integral closure of D, X be an indeterminate over D, and $N_v=\{f{\in}D[X]{\mid}c(f)_v=D\}$. In this paper, we introduce the concept of $t$-locally APVD. We show that D is a $t$-locally APVD and a UMT-domain if and only if D is a $t$-locally APVD and $D^w$ is a $PvMD$, if and only if D[X] is a $t$-locally APVD, if and only if $D[X]_{N_v}$ is a locally APVD.

• 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.

• Korean Journal of Mathematics
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• v.19 no.2
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• pp.163-169
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• 2011

Let D be an integral domain, S be a multiplicative subset of D such that DS is a PID, and D[X] be the polynomial ring over D. We show that S is an almost splitting set in D if and only if every nonzero prime ideal of D disjoint from S contains a primary element. We use this result to give a simple proof of the known result that D is a UMT-domain and Cl(D[X]) is torsion if and only if each upper to zero in D[X] contains a primary element.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.13 no.4
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• pp.109-127
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• 2003

In this paper, we analyze the network architecture, authentication, and key agreement of GSM and UMTS to compare the handover between the systems. And then, we divide authentication and key agreement procedure of mobile subscribers into several cases and finally analyze the key agreement procedure when a handover occurs in a CS-Domain and a PS-Domain.