Right chain domains with prescribed value holoid
type: Item
instance of: scholarly article
title: Right chain domains with prescribed value holoid
zbMATH Open document ID: 0837.16039
author: H. H. Brungs
author: Günter Törner
published in: Journal of Algebra
publication date: 1996-01-07T00:00:00Z
review text: Throughout, all rings have an identity and are associative, but are not necessarily commutative. A ring \(R\) is called a right chain domain if \(R\) has no zero-divisors and for any \(a,b\in R\) either \(aR\subseteq bR\) or \(bR\subsetneq aR\). A ring \(R\) is called right invariant if \(Ra\subseteq aR\) for all \(a\in R\). It is clear that the nonzero principal right ideals of \(R\) form a semigroup \(H(R)\) (the value semigroup of \(R\)) under multiplication if and only if \(R\) is right invariant. A semigroup \(H\) with identity \(e\) and a (total) order relation \(\leq\) is called an ordered semigroup if \(a\leq b\) implies \(ac\leq bc\), \(ca\leq cb\) for elements \(a,b,c\in H\). If, in addition, \(a<b\) holds if and only if \(b=ac\) for \(e\neq c\), then \(H\) is called a holoid. In this paper, the authors get the following important result: Let \(G_i\), \(i\in I\), be a set of ordered groups \(G_i\) for a (totally) ordered index set \(I\) and let \(N_i\) be a convex subset of \(G_i\) which contains the positive cone \(G^+_i\) of \(G_i\) for each \(i\). Then there exists a right invariant right chain domain \(R\) such that the associated value semigroup \(H(R)\) is isomorphic to the split holoid \(H=H_{i\in I}(G_i,N_i)\) of a family of ordered groups \(G_i\). The ring \(R\) is obtained as a suitable subring of the generalized power series ring over the wreath product of the \(G_i\). The rings \(R\) constructed in the above result are local right Bezout domains.
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16U10
Mathematics Subject Classification ID: 16K40
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 16U30
zbMATH DE Number: 815914
zbMATH Keywords: right chain domains
zbMATH Keywords: principal right ideals
zbMATH Keywords: value semigroups
zbMATH Keywords: holoids
zbMATH Keywords: ordered groups
zbMATH Keywords: generalized power series rings
zbMATH Keywords: wreath products
zbMATH Keywords: local right Bezout domains
reviewed by: Jule Zhang
MaRDI profile type: Publication
full work available at URL: jabr.1995.1248
OpenAlex ID: W2069814269
DOI: 10.1006/JABR.1995.1248
description: scientific article; zbMATH DE number 815914