Kaplansky-type Theorems, II

Abstract

Let D be an integral domain with quotient field K, X be an indeterminate over D, and D[X] be the polynomial ring over D. A prime ideal Q of D[X] is called an upper to zero in D[X] if Q = f K[X] ∩ D[X] for some f ∈ D[X]. In this paper, we study integral domains D such that every upper to zero in D[X] contains a prime element (resp., a primary element, a t-invertible primary ideal, an invertible primary ideal).

Kaplansky-Type Theorems in Graded Integral Domains

“…This 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.…”

“…This is a continuation of our works on Kaplansky-type theorems [13,20]. 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 [19,Theorem 5].…”

“…The purpose of this paper is to give several characterizations like this for graded GCD domains, graded valuation domains, graded PvMDs, etc. which can be considered as generalizations of the results in [13,20]. Precisely, in Section 1, we show that (i) a graded domain R is a (graded) GCD domain if and only if every nonzero homogeneous prime ideal of R contains a homogeneous extractor; (ii) R is a graded valuation domain if and only if every nonzero homogeneous prime ideal of R contains a homogeneous comparable element; (iii) R is a graded G-GCD domain if and only if every nonzero homogeneous prime ideal contains a d-locally extractor; (iv) R is a graded PvMD if and only if every nonzero homogeneous prime ideal contains a t-locally extractor, if and only if each nonzero homogeneous prime ideal of R contains an h-t-locally comparable element, if and only if each nonzero homogeneous prime ideal of R contains an h-t-valuation element; (v) R is a graded Krull domain if and only if every nonzero homogeneous prime ideal of R contains a homogeneous Krull element; (vi) R is a graded UFD (resp., graded π-domain, graded Krull domain) if and only if every homogeneous prime t-ideal contains a homogeneous principal (resp., invertible, t-invertible) prime ideal.…”

“…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,…”

“…Let For f ∈ D[X], 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, a prime (resp., primary) element f with c(f ) = D, and an invertible (resp., a t-invertible) primary ideal.…”