A new class of unique product monoids with applications to ring theory.
type: Item
instance of: scholarly article
title: A new class of unique product monoids with applications to ring theory.
zbMATH Open document ID: 1177.16030
author: Greg Marks
author: Ryszard Mazurek
author: Michał Ziembowski
published in: Semigroup Forum
publication date: 2009-04-27T00:00:00Z
review text: Let \((S,\leq)\) be an ordered monoid. This means that \(\leq\) is a partial order on \(S\) which is compatible with the operation on \(S\). The notion of an Artinian narrow unique product monoid is introduced. Namely, \((S,\leq)\) is such a monoid if for every two Artinian and narrow subsets \(A,B\) of \(S\) there exists an element in the product \(AB\) which is uniquely presented in the form \(ab\), where \(a\in A\) and \(b\in B\). Such monoids are unique product monoids in the standard sense, that originated from the zero divisor problems for group rings and semigroup rings. On the other hand, if \(\leq\) is the trivial order on a monoid \(S\), then \(S\) is a unique product monoid if \((S,\leq)\) is an Artinian narrow unique product monoid. Examples explaining relations between several possible definitions of unique product monoids are given. Conditions for a skew generalized power series ring \(R[\![S,\omega]\!]\), where \(S\) is an Artinian narrow unique product monoid and \(\omega\colon S\to\text{End}(R)\) is a homomorphism, to be a domain or a reduced ring are also obtained. Here skew generalized power series rings are considered in the sense introduced by the second and the third author [in Commun. Algebra 36, No. 5, 1855-1868 (2008; Zbl 1159.16032)].
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 20M25
zbMATH DE Number: 5547481
zbMATH Keywords: Artinian narrow unique product monoids
zbMATH Keywords: skew generalized power series rings
zbMATH Keywords: reduced rings
zbMATH Keywords: Armendariz rings
reviewed by: Jan Okniński
MaRDI profile type: Publication
full work available at URL: s00233-008-9063-7
OpenAlex ID: W2006423737
cites work: A note on extensions of Baer and P. P. -rings
cites work: Triangular matrix representations of ring extensions.
cites work: Q3342772
cites work: Well quasi ordering finite posets and formal languages
cites work: Ordering by Divisibility in Abstract Algebras
cites work: Ore extensions of Baer and p.p.-rings.
cites work: On Skew Armendariz Rings
cites work: Power Series Rings Satisfying a Zero Divisor Property
cites work: Q4714790
cites work: The theory of well-quasi-ordering: a frequently discovered concept
cites work: Q3999650
cites work: ARMENDARIZ RINGS RELATIVE TO A MONOID
cites work: Special Properties of Rings of Generalized Power Series
cites work: A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS
cites work: On Von Neumann Regular Rings of Skew Generalized Power Series
cites work: Q3988085
cites work: Q4145843
cites work: Armendariz rings
cites work: Noetherian rings of generalized power series
cites work: Semisimple rings and von Neumann regular rings of generalized power series
cites work: A note on u.p. groups
DOI: 10.1007/S00233-008-9063-7
Recommended article: REVERSIBLE SKEW GENERALIZED POWER SERIES RINGS
Recommended article: Some results on skew generalized power series rings
Recommended article: Uniserial rings of skew generalized power series.
Recommended article: Semisimple rings and von Neumann regular rings of generalized power series
Recommended article: Rings of generalized power series. II: Units and zero-divisors
Recommended article: Special properties of generalized power series
Recommended article: Some Ring Structures of Skew Genaralized Power Series Rings
Recommended article: Some characterizations of 2-primal skew generalized power series rings
Recommended article: Weak dimension and right distributivity of skew generalized power series rings.
Recommended article: Noetherian Generalized Power Series Rings
description: scientific article; zbMATH DE number 5547481