nLab
radical

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

$$\sqrt{I}=\{r\in R\mid \exists n\in \mathbb{N},r^n\in I\}$$\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 $\sigma :{}_R\mathrm{Mod}\to {}_R\mathrm{Mod}$ 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

$$\sigma \sigma M=\sigma \mathrm{Ker}(M\to M/\sigma M)=\mathrm{Ker}(\sigma M\to \sigma (M/\sigma M))=\mathrm{Ker}(\sigma M\to M/\sigma M)=\sigma M$$\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) $\sigma :{}_R\mathrm{Mod}\to {}_R\mathrm{Mod}$ 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 $\sigma M=\{m\in M\mid existn,I^{n}M=0\}$; 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.