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

$\sigma \sigma M = \sigma Ker(M\to M/\sigma M) = Ker (\sigma M\to \sigma(M/\sigma M)) = Ker(\sigma M\to M/\sigma M) = \sigma M$

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

- Pascual Jara, Alain Verschoren, Conchi Vidal,
*Localization and sheaves: a relative point of view*, Pitman Research Notes in Mathematics Series, 339. Longman, Harlow, 1995. xiv+235 pp. - J. L. Bueso, P. Jara, A. Verschoren,
*Compatibility, stability, and sheaves*, Monographs and Textbooks in Pure and Applied Mathematics, 185. Marcel Dekker, Inc., New York, 1995. xiv+265 pp.