On the nilpotence of the prime radical in module categories

For M∈R-Mod and τ a hereditary torsion theory on the category σ[M] we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of τ-pure prime radical Nτ(M)=Nτ as the intersection of all τ-pure prime submodules of M. We give necessary and sufficient conditions for the τ-nilpotence of Nτ(M). We prove that if M is a finitely generated R-module, progenerator in σ[M] and χ≠τ is FIS-invariant torsion theory such that M has τ-Krull dimension, then Nτ is τ-nilpotent.