Dedekind-Mertens lemma and content formulas in power series rings
type: Item
instance of: scholarly article
title: Dedekind-Mertens lemma and content formulas in power series rings
zbMATH Open document ID: 1388.13010
author: Mi Hee Park
author: B. G. Kang
author: Phan Thanh Toan
published in: Journal of Pure and Applied Algebra
publication date: 2018-04-12T00:00:00Z
review text: Let \(R\) be a commutative ring with identity. For a polynomial \(f\in R[X]\), the ideal of \(R\) generated by the coefficients of \(f\) is denoted by \(A_f\) and called the content of \(f\). The best form of the Dedekind-Mertens lemma states that if \(f,g\in R[X]\), then \(A_f^{k+1}A_g=A_f^kA_{fg}\) where \(k+1\) is the locally minimal number of generators of \(A_g\). In particular, \(A_{fg}=A_fA_g\) for all \(f,g\in R[X]\) if and only if \(R\) is a Prüfer domain and \((A_{fg})_v=(A_fA_g)_v\) for all nonzero \(f,g\in R[X]\) if and only if \(R\) is integrally closed, where \(v\) is the \(v-\)operation on \(R\). For \(f\in RX\) a power series, we define the content of \(f\) as the ideal of \(R\) generated by the coefficients of \(f\). in the present paper, the authors show that if \(R\) is a Prüfer domain then \((A_fA_g)^2=(A_fA_g)A_{fg}\) for all \(f,g\in RX\). In the particular case where \(A_g\) is locally finitely generated, we have \(A^2_fA_g=A_fA_{fg}\) for all \(f\in RX\). As a consequence, an integral domain \(R\) is completely integrally closed if and only if \((A_fA_g)_v=(A _{fg})_v\) for all nonzero \(f,g\in RX\). Other related questions are answered.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 6859161
zbMATH Keywords: power series
zbMATH Keywords: Prüfer domain
zbMATH Keywords: Dedekind-Mertens lemma
zbMATH Keywords: completely integrally closed domain.
MaRDI profile type: Publication
full work available at URL: j.jpaa.2017.09.013
OpenAlex ID: W2755862907
Wikidata QID: Q124802919
cites work: Content formulas for polynomials and power series and complete integral closure
cites work: On the Contents of Polynomials
cites work: The Dedekind-Mertens formula and determinantal rings
cites work: A generalized Dedekind-Mertens lemma and its converse
cites work: A Dedekind-Mertens theorem for power series rings
cites work: Q5687544
cites work: Some Applications of the Hilfssatz von Dedekind Mertens.
cites work: Q5661989
cites work: Q4067195
cites work: Q3205350
cites work: The Dedekind-Mertens Lemma and the contents of polynomials
cites work: Constructing chains of primes in power series rings
cites work: Q3287307
cites work: On the cardinality of star operations on a pseudo-valuation domain
cites work: Idéaux divisoriels d'un anneau de polynômes
DOI: 10.1016/J.JPAA.2017.09.013
description: scientific article; zbMATH DE number 6859161