On the nilpotence of the prime radical in module categories

Abstract

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.