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单词 ARingModuloItsJacobsonRadicalIsSemiprimitive
释义

a ring modulo its Jacobson radical is semiprimitive


Let R be a ring. Then J(R/J(R))=(0).

Proof:
We will only prove this in the case where R is a unital ring(although it is true without this assumptionPlanetmathPlanetmath).

Let [u]J(R/J(R)).By one of the characterizationsMathworldPlanetmath of the Jacobson radicalMathworldPlanetmath,1-[r][u] is left invertible for all rR,so there exists vR such that [v](1-[r][u])=1.

Then v(1-ru)=1-a for some aJ(R).There is a wR such that w(1-a)=1,and we have wv(1-ru)=1.

Since this holds for all rR,it follows that uJ(R), and therefore [u]=0.

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更新时间:2025/5/4 7:29:13