In noncommutative ring theory, particularly in the subject of noncommutative localization of rings, a kernel functor is any left exact additive subfunctor of the identity functor on the category of left modules over a ring . There is a bijective correspondence between kernel functors and uniform filters of ideals in . A functor is idempotent if and a preradical if it is additive subfunctor of the identity and for all in . A kernel functor is said to be an idempotent kernel functor if for all in ; it is idempotent as we see by calculating
In the last step, we used that is a subfunctor of the identity, hence the compositions and coincide.
The basic reference is
which is clearly written from the point of view of a ring theorist. Unfortunately, it just creates another formalism in localization theory of the categories of modules over a ring for basically the same results as P. Gabriel succeeded by more categorical formulations in his thesis published 7 years earlier. Some of the methods from Goldman, and even more from Gabriel apply for more general Grothendieck categories.