P.I. rings and the localization at height 1 prime ideals
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)]).
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