  Let $R$ be a commutative unital ring and $a\in R.$ We introduce and study
properties of a functor $a\Gamma_{a}(-),$ called the locally nilradical on the
category of $R$-modules. $a\Gamma_{a}(-)$ 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 $a\Gamma_{a}(-)$ 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.
