Jacobson radical

The Jacobson radicalMathworldPlanetmath J(R) of a unital ring R is the intersectionMathworldPlanetmath of the annihilatorsMathworldPlanetmathPlanetmathPlanetmathPlanetmath of simple (http://planetmath.org/SimpleModule) left R-modules.

The following are alternative characterizations of the Jacobson radical J(R):

  1. 1.

    The intersection of all left primitive ideals.

  2. 2.

    The intersection of all maximal left idealsMathworldPlanetmathPlanetmath.

  3. 3.

    The set of all tR such that for all rR, 1-rt is left invertible (i.e. there exists u such that u(1-rt)=1).

  4. 4.

    The largest ideal I such that for all vI, 1-v is a unit in R.

  5. 5.

    (1) - (3) with “left” replaced by “right” and rt replaced by tr.

If R is commutativePlanetmathPlanetmathPlanetmath and finitely generatedMathworldPlanetmathPlanetmathPlanetmath, then

J(R)={xRxn=0 for some n}=Nil(R).

The Jacobson radical can also be defined for non-unital rings. To do this, we first define a binary operationMathworldPlanetmath on the ring R by xy=x+y-xy for all x,yR. Then (R,) is a monoid, and the Jacobson radical is defined to be the largest ideal I of R such that (I,) is a group. If R is unital, this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the definitions given earlier.

Title Jacobson radical
Canonical name JacobsonRadical
Date of creation 2013-03-22 12:36:11
Last modified on 2013-03-22 12:36:11
Owner yark (2760)
Last modified by yark (2760)
Numerical id 19
Author yark (2760)
Entry type Definition
Classification msc 16N20
Related topic Annihilator
Related topic RadicalOfAnIdeal
Related topic SimpleModule
Related topic Nilradical
Related topic RadicalTheory
Related topic QuasiRegularity