Symmetry and localization in Noetherian prime PI rings
type: Item
instance of: scholarly article
title: Symmetry and localization in Noetherian prime PI rings
zbMATH Open document ID: 0658.16008
DOI: 10.1016/0021-8693(88)90024-5
author: A. Braun
author: Robert B. jun. Warfield
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \(R\) be a prime Noetherian P.I. ring, and let \(T(R)\) be its trace ring (so \(T(R)\) is the subring of the quotient ring of \(R\) got by adjoining to \(R\) the coefficients of the Cayley-Hamilton polynomials of the elements of \(R\), and is a finite central extension of \(R\)). Let \(P\) and \(Q\) be prime ideals of \(R\), and say that \(P\) and \(Q\) are tr-linked if there exist prime ideals \(P_ 1\) and \(Q_ 1\) of \(T(R)\) with \(P_ 1\cap R=P\), \(Q_ 1\cap R=Q\), and \(P_ 1\) and \(Q_ 1\) having the same intersection with the centre of \(T(R)\). The main theorem of this paper gives an equivalence between the 3 statements: (i) \(P\) is right (Ore) localizable; (ii) \(P\) is left localizable; and (iii) if \(Q\) is tr-linked to \(P\), then \(Q=P\). This result is, in part, a special case of a result relating the graphs of links of \(R\), and of \(T(R)\). This aspect, and its connections to the representation theory of \(R\), have been taken further in [\textit{T. H. Lenagan} and \textit{E. Letzter}, The fundamental prime ideals of a Noetherian prime P.I. ring, Proc. Edinb. Math. Soc., II. Ser. 33, No. 1, 113--121 (1990; Zbl 0697.16016)].
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16R20
zbMATH DE Number: 4075310
zbMATH Keywords: prime Noetherian P.I. ring
zbMATH Keywords: trace ring
zbMATH Keywords: quotient ring
zbMATH Keywords: prime ideals
zbMATH Keywords: left localizable
zbMATH Keywords: graphs of links
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90024-5
OpenAlex ID: W2093021703
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description: scientific article; zbMATH DE number 4075310