Three frameworks for a general theory of factorization
type: Item
instance of: scholarly article
title: Three frameworks for a general theory of factorization
zbMATH Open document ID: 1268.13001
author: R. M. Ortiz-Albino
author: D. D. Anderson
published in: Arabian Journal of Mathematics
publication date: 2012-12-18T00:00:00Z
review text: The authors discuss three different approaches for a general theory of factorization in integral domains: \(\tau\)-factorization, reduced \(\tau\)-factorization, and \(\Gamma\)-factorization. Let \(D\) be an integral domain, \(D^*\) the set of nonzero, nonunits of \(D\), and \(\tau\) a symmetric relation on \(D^*\). For \(a \in D^*, a = \lambda a_1 \cdots a_n\), where \(\lambda\) is a unit of \(D\), \(a_i \in D^*, n \geq 1\), and \(a_i \tau a_j\) for \(i \neq j\), is called a \(\tau\)-factorization of \(a\) and \(a_i\) is said to be a \(\tau\)-factor of \(a\). In the obvious way, one defines \(\tau\)-divides, \(\tau\)-atom, and \(\tau\)-prime. For example, if \(\tau = D^* \times D^*\), this gives the usual notions of factorization, and if \(a \tau b \Leftrightarrow (a,b) = D\), it gives the co-maximal theory of factorization of McAdam and Swan. The theory of reduced \(\tau\)-factorization is similar to that of \(\tau\)-factorization, but with \(\lambda = 1\). Finally, \(\Gamma\)-factorization is even more general and includes both previous factorization theories as special cases. These theories were introduced and studied by several students of the first author in their PhD Theses at the University of Iowa: A. M. Frazier (2006), S. M. Hamon (2007), R. M. Ortiz-Albino (2008), J. Reinkoester (2010), and J. Juett (2013).
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13F15
zbMATH DE Number: 6116559
zbMATH Keywords: factorization
zbMATH Keywords: \(\tau\)-factorization
zbMATH Keywords: reduced \(\tau\)-factorization
zbMATH Keywords: \(\Gamma\)-factorization
reviewed by: David F. Anderson
MaRDI profile type: Publication
full work available at URL: s40065-012-0012-7
OpenAlex ID: W2095353286
Wikidata QID: Q59292248
cites work: Q5465406
cites work: Factorization in integral domains
cites work: On a general theory of factorization in integral domains
cites work: Q5200682
cites work: Q3137901
cites work: Two counterexamples in abstract factorization
cites work: Unique comaximal factorization
cites work: Generalized relative primeness in integral domains
DOI: 10.1007/S40065-012-0012-7
description: scientific article; zbMATH DE number 6116559