radical

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

$\sqrt{I} = \{ r\in R \,|\, \exists n\in \mathbb{N}, r^n\in I \}$

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

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

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

- 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 $\sigma : {}_R Mod\to {}_R Mod$ on objects by $\sigma M = \{ m\in M\,|\, \exist n, I^n M = 0\}$; 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.

Revised on January 24, 2016 12:52:40
by Todd Trimble
(67.81.95.215)