On the nilpotency of the Jacobson radical for Noetherian rings (Q1127964)

Let \(R\) be a right noetherian ring with Jacobson radical \(J\). Using an idea of M. Prest, the author defines transfinite powers \(I^\alpha\) of an ideal \(I\) as follows: If \(\alpha=n\) is a finite ordinal, \(I^\alpha\) is the usual \(n\)-th power of \(I\), if \(\alpha\) is a limit ordinal, \(I^\alpha=\bigcap_{\gamma<\alpha} I^\gamma\), and if \(\alpha=\beta+n\) for a limit ordinal \(\beta\geq\omega\) and a natural number \(n\geq 1\), then \(I^\alpha=(I^\beta)^{n+1}\). Letting \(\kappa\) denote the Gabriel dimension of \(R\) (that is, the Krull dimension in the sense of \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)]), it is proved that \(J^{\omega\kappa+n}=0\) for some natural number \(n\). A similar result was obtained by \textit{T. H. Lenagan} and the reviewer [Commun. Algebra 7, 1-8 (1979; Zbl 0405.16008)], using a different type of transfinite power of \(J\), denoted by \(J_\alpha\). They showed that \(J_\kappa\) is nilpotent for any ring \(R\) with right Krull dimension \(\kappa\) (in the sense of Gabriel and Rentschler).