nLab
radical

Contents

Defintion

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

I={rR|n,r nI} \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 00 ideal.

For a noncommutative ring or an [[associative algebra9] 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.

Properties

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. See Shulgeifer 60

A functor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod is idempotent if σσ=σ\sigma\sigma = \sigma and a (pre)radical functor if it is an additive subfunctor of the identity functor and σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_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

σσM=σKer(MM/σM)=Ker(σMσ(M/σM))=Ker(σMM/σM)=σM \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 σ\sigma is a subfunctor of the identity, hence the compositions σMMM/σM\sigma M\hookrightarrow M\to M/\sigma M and σMσ(M/σM)M/σM\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) σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod such that σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_R Mod.

See Bueso-Jara-Verschoren 95

Examples

Example Bueso-Jara-Verschoren 95 2.3.4: Let II be a two-sided ideal in a ring and MM a left RR-module. Define the functor σ: RMod RMod\sigma : {}_R Mod\to {}_R Mod on objects by σM={mM|existn,I nM=0}\sigma M = \{ m\in M\,|\, \exist n, I^n M = 0\}; it is left exact and idempotent. If II is finitely generated as left RR-ideal (i.e. as a left RR-submodule of RR) then II is a left exact radical functor. It is clear that the formula for σM\sigma M reminds the definition of the radical of an ideal of a commutative ring.

Nonexample: the subfunctor of identity which to any module MM 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.

Revised on April 28, 2017 04:27:05 by Urs Schreiber (92.218.150.85)