properties of the Jacobson radical

Let R,T be rings and φ:RT be a surjectivePlanetmathPlanetmath homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Then φ(J(R))J(T).

We shall use the characterization of the Jacobson radicalMathworldPlanetmath as the set of all aR such that for all rR, 1-ra is left invertible.

Let aJ(R),tT. We claim that 1-tφ(a) is left invertible:

Since φ is surjective, t=φ(r) for some rR. Since aJ(R), we know 1-ra is left invertible, so there exists uR such that u(1-ra)=1. Then we have


So φ(a)J(T) as required.

Let R,T be rings. Then J(R×T)J(R)×J(T).

Let π1:R×TR be a (surjective) projectionPlanetmathPlanetmath. By the previous theorem, π1(J(R×T))J(R).

Similarly let π2:R×TT be a (surjective) projection. We see that π2(J(R×T))J(T).

Now take (a,b)J(R×T). Note that a=π1(a,b)J(R) and b=π2(a,b)J(T). Hence (a,b)J(R)×J(T) as required.

Title properties of the Jacobson radical
Canonical name PropertiesOfTheJacobsonRadical
Date of creation 2013-03-22 12:49:43
Last modified on 2013-03-22 12:49:43
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Result
Classification msc 16N20