radical

For a *commutative* ring one defines a **radical $\sqrt{I}$ of an ideal** $I\subset R$ as an ideal

$$\sqrt{I}=\{r\in R\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}\exists n\in \mathbb{N},{r}^{n}\in I\}$$

Nilradical of a commutative ring is the radical of the $0$ 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.

- E. G. Shulʹgeĭfer (Е. Г. Шульгейфер),
*К общей теории радикалов в категориях*, Матем. сб., 51(93):4 (1960), 487–500 pdf

A functor $\sigma :{}_{R}\mathrm{Mod}\to {}_{R}\mathrm{Mod}$ is **idempotent** if $\sigma \sigma =\sigma $ and a *(pre)radical functor* if it is additive subfunctor of the identity functor and $\sigma (M/\sigma (M))=0$ for all $M$ in ${}_{R}\mathrm{Mod}$ (preradical versus radical depends on an author, weather the left exactness is included or not in the definition of a radical functor). According to Goldman 1969, left exact preradical is called an **idempotent kernel functor**. It is idempotent by the calculation

$$\sigma \sigma M=\sigma \mathrm{Ker}(M\to M/\sigma M)=\mathrm{Ker}(\sigma M\to \sigma (M/\sigma M))=\mathrm{Ker}(\sigma M\to M/\sigma M)=\sigma M$$

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. In an alternative terminology, an idempotent kernel functor is any kernel functor (= left exact additive subfunctor of the identity functor) $\sigma :{}_{R}\mathrm{Mod}\to {}_{R}\mathrm{Mod}$ such that $\sigma (M/\sigma (M))=0$ for all $M$ in ${}_{R}\mathrm{Mod}$.

- 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.

Example (Bueso et al. 2.3.4): Let $I$ be a two-sided ideal in a ring and $M$ a left $R$-module. Define the functor $\sigma :{}_{R}\mathrm{Mod}\to {}_{R}\mathrm{Mod}$ ob objects by $\sigma M=\{m\in M\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}existn,{I}^{n}M=0\}$; it is left exact and idempotent. If $I$ is finitely generated as left $R$-ideal (i.e. as a left $R$-submodule of $R$) then $I$ is a left exact radical functor. It is clear that the formula for $\sigma M$ reminds the definition of the radical of an ideal of a commutative ring.

Nonexample: the subfunctor of identity which to any module $M$ assigns its socle is left exact but not a radical functor.

Created on June 8, 2011 19:02:58
by Zoran Škoda
(161.53.130.104)