Reduced modules relative to functors

Abstract

Reduced rings form an important class of rings which properly contains domains. The notion of reduced rings was studied and later extended to modules. Reduced modules have been studied using an element-wise approach. We describe the structure of reduced modules and their dual using an approach from Homological algebra. In particular, we introduce and study properties of functors that describe reduced modules defined over rings. Since reduced modules are categorical and dualisable, we study their dual known as coreduced modules. Functors that describe coreduced modules and introduced and their properties studied. As such, we reveal (resp. retrieve) new (resp. known) properties about reduced and coreduced modules over rings. For instance, reduced modules simplify the computations of local cohomology defined over a Noetherian ring. Moreover, reduced modules transform the Frobenius functor defined over a Noetherian local ring of prime characteristic p into an exact functor on a category of R-modules. Regular rings characterize coreduced modules while universally reduced modules are characterised by t-regular rings. I-coreduced modules transform the I-adic completion functor into a right exact functor on a category of R-modules.