# semiprime ideal

Let $R$ be a ring.
An ideal $I$ of $R$ is a semiprime ideal^{}
if it satisfies the following equivalent^{} conditions:

(a) $I$ can be expressed as an intersection^{} of prime ideals^{} of $R$;

(b) if $x\in R$, and $xRx\subset I$, then $x\in I$;

(c) if $J$ is a two-sided ideal^{} of $R$ and ${J}^{2}\subset I$, then $J\subset I$ as well;

(d) if $J$ is a left ideal^{} of $R$ and ${J}^{2}\subset I$, then $J\subset I$ as well;

(e) if $J$ is a right ideal of $R$ and ${J}^{2}\subset I$, then $J\subset I$ as well.

Here ${J}^{2}$ is the product of ideals $J\cdot J$.

The ring $R$ itself satisfies all of these conditions (including being expressed as an intersection of an empty family of prime ideals) and is thus semiprime.

A ring $R$ is said to be a semiprime ring if its zero ideal^{} is a semiprime ideal.

Note that an ideal $I$ of $R$ is semiprime if and only if the quotient ring^{} $R/I$ is a semiprime ring.

Title | semiprime ideal |
---|---|

Canonical name | SemiprimeIdeal |

Date of creation | 2013-03-22 12:01:23 |

Last modified on | 2013-03-22 12:01:23 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 11 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16D25 |

Related topic | NSystem |

Defines | semiprime ring |

Defines | semiprime |