On the nilpotence of the prime radical in module categories

For $M\in R$-Mod and $\tau$ a hereditary torsion theory on the category $\sigma [M]$ we use the concept of prime and semiprime module defined by Raggi et al. to introduce the concept of $\tau$-pure prime radical $\mathfrak{N}_{\tau}(M) =\mathfrak{N}_{\tau}$ as the intersection of all $\tau$-pure prime submodules of $M$. We give necessary and sufficient conditions for the $\tau$-nilpotence of $\mathfrak{N}_{\tau}(M) $. We prove that if $M$ is a finitely generated $R$-module, progenerator in $\sigma [M]$ and $\chi\neq \tau$ is FIS-invariant torsion theory such that $M$ has $\tau$-Krull dimension, then $\mathfrak{N}_{\tau}$ is $\tau$-nilpotent.