examples of semiprimitive rings

Examples of semiprimitive rings:

The integers $\mathrm{Z}$:
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 $M_n\mathbf{}\mathrm{(}D\mathrm{)}$ 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 $R\mathbf{}\mathrm{[}x\mathrm{]}$ over an integral domain $R$:
Take $a\in J(R[x])$ with $a\ne 0$. Then $ax\in J(R[x])$, since $J(R[x])$ is an ideal, and $\mathrm{deg}(ax)\ge 1$. By one of the alternate characterizations of the Jacobson radical, $1-ax$ is a unit. But $\mathrm{deg}(1-ax)=\max \{\mathrm{deg}(1),\mathrm{deg}(ax)\}\ge 1$. So $1-ax$ is not a unit, and by this contradiction 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