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Bull. Korean Math. Soc. 2015; 52(4): 1253-1268

Printed July 31, 2015

https://doi.org/10.4134/BKMS.2015.52.4.1253

Copyright © The Korean Mathematical Society.

Kaplansky-type theorems in graded integral domains

Gyu Whan Chang, Hwankoo Kim, and Dong Yeol Oh

Incheon National University, Hoseo University, Chosun University

Abstract

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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 P$v$MD (resp., G-GCD domain, GCD domain, B\'ezout domain, valuation domain, Krull domain, $\pi$-domain).

Keywords: Kaplansky-type theorem, upper to zero, prime (primary) element, graded P$v$MD, graded GCD domain, graded G-GCD domain, graded B\'ezout domain, graded valuation domain, graded Krull domain, graded $\pi$-domain

MSC numbers: 13A02, 13A15, 13F05, 13G05

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