• 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].

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1253-1268
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• 2015

It is well known that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero principal prime. This is the so-called Kaplansky's theorem. In this paper, we give this type of characterizations of a graded PvMD (resp., G-GCD domain, GCD domain, $B{\acute{e}}zout$ domain, valuation domain, Krull domain, ${\pi}$-domain).

• Bulletin of the Korean Mathematical Society
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• v.41 no.3
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• pp.473-482
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• 2004

Let D be an integral domain, I a proper ideal of D, and R =D[It, $t^{-1}$] a generalized Rees ring, where t is an indeterminate. For suitable conditions, we show that R satisfies the ACCP (resp., is a BFD, an FFD, a (pre-) Schreier domain, a G-GCD domain, a PVMD, a v-domain) if and only if D satisfies the ACCP (resp., is a BFD, an FFD, a (pre-) Schreier domain, a G-GCD domain, a PVMD, a v-domain).