Let R be a commutative unital ring and a ∈ R. We introduce and study properties of a functor aΓ a (−), called the locally nilradical on the category of R-modules. aΓ a (−) 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 aΓ a (−) 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.
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