Formal power series rings over a \(\pi \)-domain
type: Item
instance of: scholarly article
title: Formal power series rings over a \(\pi \)-domain
zbMATH Open document ID: 1181.13022
DOI: 10.4171/JEMS/186
author: B. G. Kang
author: Dong Yeol Oh
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2009-12-01T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_abstract.php
review text: Let \(R\) be an integral domain, \( \chi=\{X_{\lambda}\}_{\lambda\in \Lambda}\) an infinite set of indeterminates over \(R\) and \(S\) the weak direct sum of the additive semigroup \({\mathbb N}\) with itself \(|\Lambda|\) times. Let \(R\chi_3\) be the full ring of formal power series in \(\chi\) over \(R\), that is the set of maps \(f:S\rightarrow R\), with obvious addition and \((fg)(s)=\sum_{t+u=s} f(t)f(u)\), \(s\in S\), where \(t,u\) run in \(S\) such that \(t+u=s\). It shows that \(\mathrm{Pic } R\chi_3\cong\mathrm{Pic }R\). An integral domain is a \(\pi\)-domain if every principal ideal is a product of prime ideals, this happens exactly when it is locally an UFD and a Krull domain. It shows that \(R\chi_3\) is a \(\pi\)-domain if \(R[X_1,\ldots,X_n]\) is a \(\pi\)-domain for every \(n\geq 1\). Thus \(R\chi_3\) is a \(\pi\)-domain if \(R\) is a noetherian regular domain. Next these results are extended for non-domains.
Mathematics Subject Classification ID: 13J05
Mathematics Subject Classification ID: 13C20
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5641379
zbMATH Keywords: Krull domain
zbMATH Keywords: \(\pi\)-domain
zbMATH Keywords: unique factorization domain
zbMATH Keywords: Picard group
zbMATH Keywords: class group
MaRDI profile type: Publication
cites work: π-domains, overrings, and divisorial ideals
cites work: Globalization of Some Local Properties in Krull Domains
cites work: UNIQUE FACTORIZATION IN REGULAR LOCAL RINGS
cites work: Q3933092
cites work: Formal power series
cites work: Note generalizing a result of Samuel's
cites work: Unique factorization in power series rings and semigroups
cites work: Q5670698
cites work: Power series rings over a Krull domain
cites work: Q5661989
cites work: On the converse of a well-known fact about Krull domains
cites work: Q4049183
cites work: A Characterization of General Z.P.I.-Rings
cites work: Q5777228
cites work: On the unique factorization theorem for formal power series
cites work: On unique factorization domains
OpenAlex ID: W2031295477
description: scientific article; zbMATH DE number 5641379