# 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 $\mathrm{deg}(1-ax)=\mathrm{max}\{\mathrm{deg}(1),\mathrm{deg}(ax)\}\ge 1$.
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 |