Construction of left fir monoid rings
type: Item
instance of: scholarly article
title: Construction of left fir monoid rings
zbMATH Open document ID: 0804.16012
author: Ferran Cedó
author: Andreu Pitarch
published in: Journal of Algebra
publication date: 1995-01-12T00:00:00Z
review text: In any monoid \(M\) (always with cancellation) define a preordering by putting \(a\leq b\) if \(b=ad\) for some \(d\in M\); this defines a partial ordering of classes of right associated elements of \(M\). \textit{I. B. Kozhukhov} [Algebra Logika 21, 37-59 (1982; Zbl 0512.16004)] has shown that for any ring \(K\) the monoid ring \(KM\) is a left fir if and only if \(K\) is a skew field, \(M\) has cancellation and the above partial order is a well-ordering on the left factors of any element, the unit group \(U(M)\) of \(M\) is free and \(M\) is irreflexive i.e. \(b=abc\) for a non-unit \(b\) implies \(a=c=1\). Such an \(M\) is also called a left fir monoid. For any element \(c\) of \(M\) the set of left factors (up to right associates) forms an ordinal \(\tau(c)\) and if \(\tau(M)=\sup\{\tau(c)\mid c\in M\}\), Kozhukhov proposed an example, for any ordinal \(\alpha\), of a left fir monoid \(M\) with \(\tau(M)=\alpha\). However his construction contains a mistake. The authors make a careful analysis of left fir monoids which results in a precise description of left fir monoids \(M\) with a given \(\tau(M)\). In particular this leads to a construction of a left fir monoid \(M\) with a given free group as group of units and a given ordinal as value of \(\tau(M)\).
Mathematics Subject Classification ID: 16E60
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 16U60
zbMATH DE Number: 589966
zbMATH Keywords: partial ordering
zbMATH Keywords: right associated elements
zbMATH Keywords: monoid ring
zbMATH Keywords: cancellation
zbMATH Keywords: unit group
zbMATH Keywords: left fir monoid
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: jabr.1994.1137
OpenAlex ID: W1993606469
DOI: 10.1006/JABR.1994.1137
Recommended article: Monoid rings that are firs
Recommended article: On rigid monoids and 2-fir monoid rings
Recommended article: On monoids, 2-firs, and semifirs.
Recommended article: Right-orderability versus left-orderability for monoids
Recommended article: On semifir monoid rings
Recommended article: A question of cohn on semifir monoid rings
Recommended article: Les f-réguliers à gauche
Recommended article: F-Monoids
Recommended article: Homological finiteness properties of monoids, their ideals and maximal subgroups.
Recommended article: Codes limites et factorisations finies du monoïde libre
description: scientific article; zbMATH DE number 589966