# examples of semiprimitive rings

The integers $\mathbb{Z}$:
Since $\mathbb{Z}$ is commutative, any left ideal is two-sided. So the maximal left ideals of $\mathbb{Z}$ are the maximal ideals of $\mathbb{Z}$, which are the ideals $p\mathbb{Z}$ for $p$ prime. So $J(\mathbb{Z})=\bigcap_{p}p\mathbb{Z}=(0)$, as there are infinitely many primes.

A matrix ring $M_{n}(D)$ over a division ring $D$:
The ring $M_{n}(D)$ is simple, so the only proper ideal is $(0)$. Thus $J(M_{n}(D))=(0)$.

A polynomial ring $R[x]$ over an integral domain $R$:
Take $a\in J(R[x])$ with $a\neq 0$. Then $ax\in J(R[x])$, since $J(R[x])$ is an ideal, and $\deg(ax)\geq 1$. By one of the alternate characterizations of the Jacobson radical, $1-ax$ is a unit. But $\deg(1-ax)=\max\{\deg(1),\deg(ax)\}\geq 1$. So $1-ax$ is not a unit, and by this contradiction we see that $J(R[x])=(0)$.

Title examples of semiprimitive rings ExamplesOfSemiprimitiveRings 2013-03-22 12:50:39 2013-03-22 12:50:39 yark (2760) yark (2760) 12 yark (2760) Example msc 16N20