For a commutative ring one defines a radical of an ideal as an ideal
Nilradical of a commutative ring is the radical of the ideal.
For a noncommutative ring or an associative algebra there are many competing notions of a radical of a ring like Jacobson radical, Levitzky radical; and sometimes of radicals of ideals or, more often arbitrary modules of a ring.
Some of the abstract properties of such functors are used in noncommutative localization theory, when defining so called radical functors. The latter are generalized for arbitrary Grothendieck categories. Finally there are some notions of radicals in nonadditive categories.
A functor is idempotent if and a (pre)radical functor if it is an additive subfunctor of the identity functor and for all in (preradical versus radical depends on an author, whether the left exactness is included or not in the definition of a radical functor). According to Goldman 1969, a left exact preradical is called an idempotent kernel functor. It is idempotent by the calculation
In the last step, we used that is a subfunctor of the identity, hence the compositions and coincide. In an alternative terminology, an idempotent kernel functor is any kernel functor (= left exact additive subfunctor of the identity functor) such that for all in .
Example (Bueso et al. 2.3.4): Let be a two-sided ideal in a ring and a left -module. Define the functor on objects by ; it is left exact and idempotent. If is finitely generated as left -ideal (i.e. as a left -submodule of ) then is a left exact radical functor. It is clear that the formula for reminds the definition of the radical of an ideal of a commutative ring.
Nonexample: the subfunctor of identity which to any module assigns its socle is left exact but not a radical functor.