The Nilpotence of the τ-Closed Prime Radical in Rings with τ-Krull Dimension

Necessary and sufficient conditions are provided for the nilpotence of the intersection of all τ-closed prime ideals of a ring R with τ-Krull dimension, where τ denotes a hereditary torsion theory on the category Mod-R of right R-modules. It is shown that these conditions hold, in particular, when τ is ideal invariant. Furthermore, conditions are provided for Ass(M) to be non-empty for every τ-torsionfree right R-module M ≠ 0.