Closed polynomials in polynomial rings over integral domains
type: Item
instance of: scholarly article
title: Closed polynomials in polynomial rings over integral domains
zbMATH Open document ID: 1320.13010
author: Hideo Kojima
author: Takanori Nagamine
published in: Journal of Pure and Applied Algebra
publication date: 2015-08-31T00:00:00Z
review text: A polynomial \(f(X_1,\dots,X_n)\) over an integral domain \(R\) is called a closed polynomial, if the ring \(R[f]\) is integrally closed in \(R[X_1,\dots,X_n]\). These polynomials have been studied earlier in the case when \(R\) is a field (see e.g. [\textit{A. Nowicki}, Nagoya Math. J. 109, 151--157 (1988; Zbl 0642.13016); \textit{M. Ayad}, Acta Arith. 105, No. 1, 9--28 (2002; Zbl 1038.12001); \textit{I. V. Arzhantsev} and \textit{A. P. Petravchuk}, Ukr. Mat. Zh. 59, No. 12, 1587--1593 (2007; Zbl 1164.13302)]), as well as in the case when \(R\) is a unique factorization domain [\textit{M. Kato} and \textit{H. Kojima}, Commun. Algebra 43, No. 5, 1935--1938 (2015; Zbl 1327.13033)]. The authors treat now the case when \(R\) is an arbitrary integral domain, and generalize the results of papers quoted above, establishing in particular characterizations of closed polynomials. We quote one typical result (second part of Theorem 3.1): If \(R\) is an integral domain with quotient field \(K\) of zero characteristic, and the polynomial \(f\in R[X_1,\dots,X_n]\setminus R\) satisfies \(K[f]\cap R[X_1,\dots,X_n] = R[f]\), then \(f\) is closed if and only if there exists an \(R\)-derivation \(d\) on \(R[X_1,\dots,X_n]\) with Ker \(d = R[f]\).
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13N15
zbMATH DE Number: 6476979
zbMATH Keywords: closed polynomials
zbMATH Keywords: polynomial rings
zbMATH Keywords: higher derivations
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.05.028
OpenAlex ID: W657283344
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DOI: 10.1016/J.JPAA.2015.05.028
description: scientific article; zbMATH DE number 6476979