Localization, completion and the AR property in Noetherian P.I. rings
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.
scientific article; zbMATH DE number 997617
scientific article