Localization, completion and the AR property in Noetherian P.I. rings (Q677450)

Let \(R\) be a Noetherian ring satisfying the second layer condition, (for example, a group algebra of a polycyclic-by-finite group, an enveloping algebra of a solvable Lie algebra, or a P.I. ring), and let \(P\) be a prime ideal of \(R\). This paper addresses the relation between the Ore localization of \(P\) and the Noetherian property of the completion \(\widehat R:=\lim_{\gets i}R/P^i\). The idea that these two conditions should be related can be traced back to \textit{A. W. Goldie} [J. Algebra 5, 89-105 (1967; Zbl 0154.28801)]. The central result of the present paper is the following: Let \(R\) and \(P\) be as above, and suppose that \(P\) is semiprimitive, that it is contained in a localizable semiprime ideal, and that \(\lim_{\gets i}R/P^i\) is Noetherian. Then \(P\) is Ore localizable. The author explores consequences of this for enveloping algebras and for P.I. rings, and considers the extent to which the converse fails to be true.