On the radical of multigraded modules

We define a functor r⁎r⁎ from the category of positively determined modules to the category of squarefree modules which plays the role of passing from a monomial ideal to its radical. By using this functor, we generalize several results on properties that are shared by a monomial ideal and its radical. Moreover, we study the connection of r⁎r⁎ to the Alexander duality and Auslander–Reiten translate.