Construction of lattice orders on the semigroup ring of a positive rooted monoid
type: Item
instance of: scholarly article
title: Construction of lattice orders on the semigroup ring of a positive rooted monoid
zbMATH Open document ID: 1020.06008
DOI: 10.1016/S0021-8693(03)00053-X
author: Jingjing Ma
author: Stuart A. Steinberg
published in: Journal of Algebra
publication date: 2003-04-28T00:00:00Z
review text: Let \((M,+,\leq)\) be a strictly partially ordered monoid such that (1) the set of all upper bounds of each element in \(M\) is a chain, (2) each non-zero element \(a\in M\) is positive (i.e., \(a<a+a\) for any \(a\neq 0)\), and (3) there exists \(m\in M\), \(m\neq 0\), comparable to every non-zero element of \(M\). In this paper the monoid ring \(A(M)\) for a totally ordered domain \(A\) is studied with respect to the question of which lattice-orders exist on \(A(M)\) such that the squares of its elements are positive. Under certain conditions all such orderings are constructed, which make the monoid ring into a lattice-ordered ring. In particular, for the polynomial ring \(A[x]\) all partial orders are determined for which \(A[x]\) is not totally ordered but is a lattice-ordered ring such that every element is positive and \(x\) is a \(d\)-element. For a partially ordered ring \(R\) an element \(a\in R^+\) is called a \(d\)-element on the left lattice-ordered \(R\)-module \(A\) if \(x\wedge y=0\) implies \(ax\wedge ay=0\). Properties of \(d\)-elements are used to characterize lattice-ordered rings that are quadratic extensions of totally ordered division rings. (The number of concepts used in the paper is too high to be reproduced here in order to discuss the main results of the paper in more detail).
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 1901838
zbMATH Keywords: \(l\)-field
zbMATH Keywords: positive monoid
zbMATH Keywords: positive squares
zbMATH Keywords: partially ordered monoid
zbMATH Keywords: monoid ring
zbMATH Keywords: totally ordered domain
zbMATH Keywords: lattice-ordered ring
zbMATH Keywords: partial orders
zbMATH Keywords: partially ordered ring
zbMATH Keywords: quadratic extensions
zbMATH Keywords: division rings
reviewed by: H. Mitsch
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901838