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 ${}_R Mod$ 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

• O. Goldman, Rings and modules of quotients, J. Algebra 13, 1969 10–47, MR245608, doi

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.