# associated prime

Let $R$ be a ring,
and let $M$ be an $R$-module.
A prime ideal^{} $P$ of $R$
is an for $M$
if $P=\mathrm{ann}(X)$, the annihilator^{} of some nonzero submodule $X$ of $M$.

Note that if this is the case, then the module ${\mathrm{ann}}_{M}(P)$ contains $X$, has $P$ as its annihilator, and is a faithful (http://planetmath.org/FaithfulModule) $(R/P)$-module.

If, in addition, $P$ is equal to the annihilator of a submodule of $M$ that is a fully faithful (http://planetmath.org/FaithfulModule) $(R/P)$-module, then we call $P$ an of $M$.

Title | associated prime |
---|---|

Canonical name | AssociatedPrime |

Date of creation | 2013-03-22 12:01:37 |

Last modified on | 2013-03-22 12:01:37 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 16D25 |

Synonym | annihilator prime |