nLab
kernel functor

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 RMod{}_R Mod of left modules over a ring RR. There is a bijective correspondence between kernel functors and uniform filters of ideals in RR. A functor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod is idempotent if σσ=σ\sigma\sigma = \sigma and a preradical if it is additive subfunctor of the identity and σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_R Mod. A kernel functor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod is said to be an idempotent kernel functor if σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_R Mod; it is idempotent as we see by calculating

σσM=σKer(MM/σM)=Ker(σMσ(M/σM))=Ker(σMM/σM)=σM \sigma \sigma M = \sigma Ker(M\to M/\sigma M) = Ker (\sigma M\to \sigma(M/\sigma M)) = Ker(\sigma M\to M/\sigma M) = \sigma M

In the last step, we used that σ\sigma is a subfunctor of the identity, hence the compositions σMMM/σM\sigma M\hookrightarrow M\to M/\sigma M and σMσ(M/σM)M/σM\sigma M\to \sigma(M/\sigma M)\to M/\sigma M 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.