Journal of the
Korean Mathematical Society JKMS

Klein proved that polynomial rings over nil rings of bounded index are also nil of bounded index; while Puczy\l owski and Smoktunowicz described the nilradical of a power series ring with an indeterminate. We extend these results to those with any set of commuting indeterminates. We also study prime radicals of power series rings over some class of rings containing the case of bounded index, finding some examples which elaborate our arguments; and we prove that $R$ is a PI ring of bounded index then the power series ring $R[[X]]$, with $X$ any set of indeterminates over $R$, is also a PI ring of bounded index, obtaining the Klein's result for polynomial rings as a corollary.

Keywords: nilradical, Wedderburn radical, polynomial ring, power series ring, nil ring of bounded index

MSC numbers: 16N40, 16R20, 16S99