Kaplansky-Type Theorems in Graded Integral Domains

Abstract

Abstract. 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, Bézout domain, valuation domain, Krull domain, π-domain). IntroductionThis is a continuation of our works on Kaplansky-type theorems [13,20] Later, in [20], the second-named author gave a Kaplansky-type characterization of G-GCD domains and PvMDs and gave an ideal-wise version of Kaplansky-type theorems. This ideal-wise version is then used to give characterizations of UFDs, π-domains, and Krull domains. , let c(f ) be the ideal of D generated by the coefficients of f . In [13], the first two authors of this paper studied an integral domain D such that every upper to zero in D[X] contains a prime (resp., primary) element,

Global perinormality in a generalized D + M construction

“…) where corresponding localizations are equal and hence corresponding graded localizations are equal. Note that (R (p) ) ′ is a Krull domain by the Mori-Nagata Integral Closure Theorem and is graded because the integral closure is a graded overring by [Cha17,Lemma 1.6]. Hence…”