# primary ideal

An ideal $Q$ in a commutative ring $R$ is a *primary ideal* if for all elements $x,y\in R$, we have that if $xy\in Q$, then either $x\in Q$ or ${y}^{n}\in Q$ for some $n\in \mathbb{N}$.

This is clearly a generalization^{} of the notion of a prime ideal^{}, and (very) loosely mirrors the relationship in $\mathbb{Z}$ between prime numbers and prime powers.

It is clear that every prime ideal is primary.

Example. Let $Q=(25)$ in $R=\mathbb{Z}$. Suppose that $xy\in Q$ but $x\notin Q$. Then $25|xy$, but 25 does not divide $x$. Thus 5 must divide $y$, and thus some power of $y$ (namely, ${y}^{2}$), must be in $Q$.

The radical^{} of a primary ideal is always a prime ideal. If $P$ is the radical of the primary ideal $Q$, we say that $Q$ is *$P$-primary*.

Title | primary ideal |
---|---|

Canonical name | PrimaryIdeal |

Date of creation | 2013-03-22 14:15:01 |

Last modified on | 2013-03-22 14:15:01 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 6 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 13C99 |

Defines | primary |

Defines | $P$-primary |