examples of semiprimitive rings

The integers Z:
Since is commutativePlanetmathPlanetmathPlanetmath, any left idealMathworldPlanetmathPlanetmath is two-sided. So the maximal left ideals of are the maximal idealsMathworldPlanetmath of , which are the ideals p for p prime. So J()=pp=(0), as there are infinitely many primes.

A matrix ring Mn(D) over a division ring D:
The ring Mn(D) is simple, so the only proper idealMathworldPlanetmath is (0). Thus J(Mn(D))=(0).

A polynomial ring R[x] over an integral domainMathworldPlanetmath R:
Take aJ(R[x]) with a0. Then axJ(R[x]), since J(R[x]) is an ideal, and deg(ax)1. By one of the alternate characterizations of the Jacobson radicalMathworldPlanetmath, 1-ax is a unit. But deg(1-ax)=max{deg(1),deg(ax)}1. So 1-ax is not a unit, and by this contradictionMathworldPlanetmathPlanetmath we see that J(R[x])=(0).

Title examples of semiprimitive rings
Canonical name ExamplesOfSemiprimitiveRings
Date of creation 2013-03-22 12:50:39
Last modified on 2013-03-22 12:50:39
Owner yark (2760)
Last modified by yark (2760)
Numerical id 12
Author yark (2760)
Entry type Example
Classification msc 16N20