Reduced and coreduced modules with respect to inverse families of ideals

Let $R$ be a commutative ring and $Φ$ an inverse family of ideals with greatest proper member $\mathcal{L}$. We introduce and study $Φ$-reduced and $Φ$-coreduced modules, extending the corresponding notions for powers of a single ideal. We show that, on these classes of modules, the generalized tors