nLab
radical

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Definition

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 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.

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 RMod{ }_R Mod (left modules over a ring RR, 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 RMod{ }_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:σ1 RModi: \sigma \hookrightarrow 1_{ _R Mod} of the identity on RMod{ }_R Mod, the additive category of left RR-modules. Naturality of ii implies the equation iσi=iiσi \circ \sigma i = i \circ i\sigma, whence σi=iσ\sigma i = i\sigma by monicity of ii. Some authors refer to these as preradical functors (e.g., Mirhosseinkhani 2010).

Such a functor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod is idempotent if σi=iσ:σσσ\sigma i = i\sigma: \sigma \to \sigma\sigma is an isomorphism, and is called a radical functor if in addition σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_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

σσM=σKer(MM/σM)=Ker(σMσ(M/σM))=Ker(σMMM/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\to M/\sigma M) = \sigma M

where 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.

However, beware that other authors call a left exact additive subfunctor σ: RMod RMod\sigma: {}_R Mod\to {}_R Mod of the identity functor a kernel functor, and then call a kernel functor σ\sigma an idempotent kernel functor if σ(M/σ(M))=0\sigma(M/\sigma(M))=0 for all MM in RMod{}_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 II be a two-sided ideal in a ring. Define a functor σ: RMod RMod\sigma : {}_R Mod\to {}_R Mod on objects by σM={mM|n,I nM=0}\sigma M = \{ m\in M\,|\, \exists 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.

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

See also

Last revised on March 5, 2018 at 14:16:38. See the history of this page for a list of all contributions to it.