P.I. rings and the localization at height 1 prime ideals
type: Item
instance of: scholarly article
title: P.I. rings and the localization at height 1 prime ideals
zbMATH Open document ID: 0658.16011
DOI: 10.1007/BF02767357
author: A. Braun
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(R\) be a prime ring satisfying a polynomial identity, and let \(P\) be a prime ideal of \(R\). In [\textit{A. Braun} and \textit{R. B. Warfield} jun., J. Algebra 118, No. 2, 322--335 (1988; Zbl 0658.16008)], a criterion is obtained (involving the finite set of prime ideals of the trace ring of \(R\) lying over \(P\)) for \(P\) to be a localized ideal, provided \(R\) is Noetherian. In the present note a similar result is obtained without the hypothesis that \(R\) is Noetherian, but assuming instead that \(R\) is a finitely generated algebra over a central Noetherian subring, and that \(P\) has height 1, with \(P/P^ 2\) a finitely generated \(R\)-module. In fact, the last of these hypotheses can be removed (see \textit{T. H. Lenagan}, Isr. J. Math. 67, No. 3, 345--352 (1989; Zbl 0739.16017)]).
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16D99
zbMATH DE Number: 4075313
zbMATH Keywords: prime ring
zbMATH Keywords: polynomial identity
zbMATH Keywords: prime ideal
zbMATH Keywords: trace ring
zbMATH Keywords: lying over
zbMATH Keywords: localized ideal
zbMATH Keywords: finitely generated algebra
zbMATH Keywords: central Noetherian subring
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Integral ring homomorphisms
cites work: An additivity principle for p.i. rings
cites work: Localization in prime Noetherian P.I. rings
cites work: Symmetry and localization in Noetherian prime PI rings
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cites work: Q3717234
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cites work: Integral extensions of rings satisfying a polynomial identity
cites work: Bezout rings and serial rings
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description: scientific article; zbMATH DE number 4075313