# primitive ideal

Let $R$ be a ring, and let $I$ be an ideal of $R$.
We say that $I$ is a left (right) primitive ideal
if there exists a simple left (right) $R$-module $X$
such that $I$ is the annihilator^{} of $X$ in $R$.

We say that $R$ is a left (right) primitive ring
if the zero ideal^{} is a left (right) primitive ideal of $R$.

Note that $I$ is a left (right) primitive ideal if and only if $R/I$ is a left (right) primitive ring.

Title | primitive ideal |
---|---|

Canonical name | PrimitiveIdeal |

Date of creation | 2013-03-22 12:01:45 |

Last modified on | 2013-03-22 12:01:45 |

Owner | antizeus (11) |

Last modified by | antizeus (11) |

Numerical id | 6 |

Author | antizeus (11) |

Entry type | Definition |

Classification | msc 16D25 |

Synonym | primitive ring |