The nilpotence of the tau-closed prime radical in rings with tau-Krull dimension

Necessary and sufficient conditions are provided for the nilpotence of the intersection of all tau-closed prime ideals of a ring R with tau-Krull dime nsion, where tau denotes a hereditary torsion theory on the category Mod-R of right R-modules. It is shown that these conditions hold, in particular, when tau is ideal invariant. Furthermore, conditions are provided for Ass(M ) to be non-empty for every tau-torsionfree right R-module M not equal 0. ( C) 2000 Academic Press.