• 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).