The locally nilradical for modules over commutative rings

Abstract

Let R be a commutative unital ring and a ∈ R. We introduce and study properties of a functor aΓa(−), called the locally nilradical on the category of R-modules. aΓa(−) is a generalisation of both the torsion functor (also called section functor) and Baer’s lower nilradical for modules. Several local-global properties of the functor aΓa(−) are established. As an application, results about reduced R-modules are obtained and hitherto unknown ring theoretic radicals as well as structural theorems are deduced.