nLab radical

Definition

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

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

An ideal is called a radical ideal if it is equal to its own radical.

The 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 such as Jacobson radical, Levitzky radical, and sometimes of radicals of ideals or, more often, of radicals of arbitrary modules of a ring.

Given a (possibly noncommutative) ring $R$ radical of an $R$-module is the intersection of all of its maximal submodules. It may also be presented as the internal sum of all its superfluous submodules. This is the notion dual to the notion of a socle.

Radical functors

Each of the notions of radical mentioned above are functorial, and some of the abstract properties of such functors are used in noncommutative localization theory, when defining so called radical functors. Classically these were considered for module categories ${ }_R Mod$ (left modules over a ring $R$, but there are generalizations for arbitrary Grothendieck categories, and there are also some notions of radical for nonadditive categories. See Shulgeifer 60.

We define here radical functors on ${ }_R Mod$, but warn that there are some terminological discrepancies across the literature.

However they are defined, all notions of radical involve additive subfunctors $i: \sigma \hookrightarrow 1_{ _R Mod}$ of the identity on ${ }_R Mod$, the additive category of left $R$-modules. Naturality of $i$ implies the equation $i \circ \sigma i = i \circ i\sigma$, whence $\sigma i = i\sigma$ by monicity of $i$. Some authors refer to these as preradical functors (e.g., Mirhosseinkhani 2010).

Such a functor $\sigma: {}_R Mod\to {}_R Mod$ is idempotent if $\sigma i = i\sigma: \sigma\sigma \to \sigma$ is an isomorphism, and is called a radical functor if in addition $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$. Note however that some authors call this a preradical functor, and define a radical functor to be such a preradical functor that is left exact.

Following Goldman 1969, a left exact additive subfunctor of the identity is called an idempotent kernel functor. Observe that such is idempotent by the calculation

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

where 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.

However, beware that other authors call a left exact additive subfunctor $\sigma: {}_R Mod\to {}_R Mod$ of the identity functor a kernel functor, and then call a kernel functor $\sigma$ an idempotent kernel functor if $\sigma(M/\sigma(M))=0$ for all $M$ in ${}_R Mod$. In other words, their idempotent kernel functors coincide with what other authors call radical functors in the strong (left exact) sense above.

See Bueso-Jara-Verschoren 95

Examples

Example (Bueso-Jara-Verschoren 95 2.3.4): Let $I$ be a two-sided ideal in a ring. Define a functor $\sigma : {}_R Mod\to {}_R Mod$ on objects by $\sigma M = \{ m\in M\,|\, \exists n, I^n M = 0\}$; 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.

References

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

• 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.

• O. Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10-47.

See also

• Wikipedia, radical of a ring, radical of a module

• M Grandis, G Janelidze, László Márki, Non-pointed exactness, radicals, closure operators, J. Austr. Math. Soc. 94:3 (2013) 348–361 doi pdf

• D. N. Dikranjan, Walter Tholen, Categorical structure of closure operators with applications to topology, Kluwer 1995 (Series Math. its Appl.)