J. Austral. Math. Soc.  72 (2002), 23-31

Radicals and polynomial rings

K. I. Beidar   Department of Mathematics   National Cheng-Kung University   Tainan   Taiwan   beidar@mail.ncku.edu.tw E. R. Puczylowski   Institute of Mathematics   University of Warsaw   Warsaw   Poland   edmundp@mimuw.edu.pl and R. Wiegandt   Institute of Mathematics   Hungarian Academy of Sciences   Budapest   Hungary   wiegandt@math-inst.hu

Abstract

We prove that polynomial rings in one indeterminate over nil rings are antiregular radical and uniformly strongly prime radical. These give some approximations of Kothe's problem. We also study the uniformly strongly prime and superprime radicals of polynomial rings in non-commuting indeterminates. Moreover, we show that the semi-uniformly strongly prime radical coincides with the uniformly strongly prime radical and that the class of semi-superprime rings is closed under taking finite subdirect sums.

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