Orderings, real places, and valuations on noncommutative integral domains
type: Item
instance of: scholarly article
title: Orderings, real places, and valuations on noncommutative integral domains
zbMATH Open document ID: 0934.16041
published in: Journal of Algebra
publication date: 2000-02-02T00:00:00Z
review text: This paper may be viewed as a continuation of previous work of the authors (jointly with \textit{K. H. Leung} [J. Algebra 198, No. 2, 412-427 (1997; Zbl 0909.16027)]) on the real spectrum of noncommutative rings, its principal aim being the development of an associated theory of real places and valuations. In particular, the relation between the set of real places on \(A\) and the set \(X\) of support \(\{0\}\) orderings on \(A\) is carefully examined, where for any ordering \(P\) of \(A\) the support of \(P\) is the prime ideal \(P\cap-P\). Given a preordering \(T\) in \(A\) (a subset \(T\subseteq A\) with \(\sum A^2(T)\subseteq T\), where \(A^2(T)\) consists of all permuted products of elements \(a_1,\ldots,a_n\), \(s_1,\ldots,s_m\), where \(a_i\in A\) and \(s_j\in T\)) with \(T\cap-T=\{0\}\), one denotes by \(X_T\) the set of support \(\{0\}\) orderings on \(A\) containing \(T\) and by \(G_T\) the set of all restrictions of elements of \(G\) to \(X_T\), where \(G\) consists of all mappings of the type \(\overline a\colon X\to\{-1,1\}\) with \(\overline a(P)=1\) resp. \(\overline a(P)=-1\) for \(a\in P\) and \(a\in-P\), for some element \(a\in A\). Each ordering \(P\in X_T\) yields a character \(\chi_P\) on \(G_T\) given by \(\overline a\mapsto\overline a(P)\). One calls \(X_T\) a fan if every character \(\chi\) with values in \(\{-1,1\}\) on \(G_T\) satisfying \(\chi(-1)=-1\) is of the form \(\chi_P\) for some ordering \(P\in X_T\). Actually, one may show that this implies any subset \(P\) of \(A\) satisfying \(T\subseteq P\), \(PP\subseteq P\), \(P\cup-P=A\) and \(P\cap-P=\{0\}\) to be an ordering (i.e., \(P\) is automatically also closed under addition). Denoting by \(M\) the set of so-called order compatible real places, one of the main results of the paper states that the canonical map \(\lambda\colon X\to M\) defines a \(P\)-structure on the space of orderings \((X,G)\), in the sense that \(\lambda^{-1}(\alpha)\) is a fan in \(X\) for every \(\alpha\in M\) a…
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16U10
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 1271650
zbMATH Keywords: fans
zbMATH Keywords: real spectra of noncommutative rings
zbMATH Keywords: real places
zbMATH Keywords: orderings
zbMATH Keywords: order compatible real places
zbMATH Keywords: integral domains
zbMATH Keywords: Bröcker's trivialization theorem
zbMATH Keywords: quantum planes
author: Murray Marshall
author: Yu-Fei Zhang
MaRDI profile type: Publication
full work available at URL: jabr.1998.7630
OpenAlex ID: W2051432937
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DOI: 10.1006/JABR.1998.7630
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description: scientific article; zbMATH DE number 1271650