Generalized power series with a limited number of factorizations
type: Item
instance of: scholarly article
title: Generalized power series with a limited number of factorizations
zbMATH Open document ID: 1506.13003
author: Ngoc P. Aylesworth
author: J. R. Juett
published in: Journal of Commutative Algebra
publication date: 2022-12-19T00:00:00Z
full work available at URL: jca.2022.14.471.full
review text: The paper deals with generalized power series ring. Let's first review it. Let \((S,+,\leq)\) be a strictly ordered commutative monoid, that is if \(s,t,u\in S\) are such that \(s<t\) then \(s+u<t+u\). A subset \(F\) of \(S\) is said Artinian if each decreasing sequence of elements of \(F\) stabilizes and is narrow if it contains no infinite set of incomparable elements. Now, let \(R\) be a commutative ring with unit. The generalized power series ring \([[R,S^{\leq}]]\) consists of all the formal sums \(\displaystyle f=\sum_{s\in S}a_sX^s\) with \(a_s\in A\) such that \(\mathrm{supp}(f)=\{s\in S; a_s\not=0\}\) is Artinian and narrow. This type of rings include (Laurent) power series and large (Laurent) polynomials. In the third section of this paper, the authors investigate when \([[R,S^{\leq}]]\) is an integral domain where every nonzero nonunit has only finitely many divisors up to associates. As it is very known, beside the context of integral domains, factorization theory is defined for commutative rings with zero divisors, where, as expected many difficulties and problems arise. In the fourth section, the authors investigate when \([[R,S^{\leq}]]\) satisfies the ``limited factorization'' properties. Many applications, examples and diagrams are given.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13F15
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 7634468
zbMATH Keywords: finite factorization ring
zbMATH Keywords: generalized power series ring
zbMATH Keywords: Krull domain
zbMATH Keywords: Cohen-Kaplansky ring
MaRDI profile type: Publication
OpenAlex ID: W4309223436
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DOI: 10.1216/JCA.2022.14.471