On rings without a certain divisibility property
type: Item
instance of: scholarly article
title: On rings without a certain divisibility property
zbMATH Open document ID: 0651.13005
DOI: 10.1016/0022-314X(88)90060-1
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: An integral domain R with field of fractions F is called D-ring if whenever \(f,g\in R[X]\) are nonzero and such that if f(a) divides g(a) for all but finitely many \(a\in R\), then f divides g in F[X]. The author studies Dedekind non-D-rings. If R is a non-D-ring and \(f\in R[X]\), then f is called unit-valued if f(R)\(\subset U(R)\) (units of R). If R is a Dedekind non-D-ring with ideal class group Cl(R) and \(f\in R[X]\) is a unit-valued polynomial, \(\deg (f)=n\geq 2\), then Cl(R) is a torsion group with exponent d for some divisor d of n. Applications to algebraic number fields are given.
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4061368
zbMATH Keywords: integral domain
zbMATH Keywords: Dedekind non-D-rings
zbMATH Keywords: ideal class group
zbMATH Keywords: unit-valued polynomial
author: K. Alan Loper
MaRDI profile type: Publication
cites work: On rings with a certain divisibility property
cites work: Number fields
cites work: Q3249822
full work available at URL: 0022-314x(88)90060-1
OpenAlex ID: W1998795721
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description: scientific article; zbMATH DE number 4061368