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单词 JacobsonRadical
释义

Jacobson radical


The Jacobson radicalMathworldPlanetmath J(R) of a unital ring R is the intersectionMathworldPlanetmathof 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 isleft 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 aunit 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 Rby 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 Rsuch that (I,) is a group.If R is unital, this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the definitions given earlier.

TitleJacobson radical
Canonical nameJacobsonRadical
Date of creation2013-03-22 12:36:11
Last modified on2013-03-22 12:36:11
Owneryark (2760)
Last modified byyark (2760)
Numerical id19
Authoryark (2760)
Entry typeDefinition
Classificationmsc 16N20
Related topicAnnihilator
Related topicRadicalOfAnIdeal
Related topicSimpleModule
Related topicNilradical
Related topicRadicalTheory
Related topicQuasiRegularity
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更新时间:2025/5/5 0:17:06