# ideals with maximal radicals are primary

Proposition^{}. Assume that $R$ is a commutative ring and $I\subseteq R$ is an ideal, such that the radical^{} $r(I)$ of $I$ is a maximal ideal^{}. Then $I$ is a primary ideal^{}.

Proof. We will show, that every zero divisor^{} in $R/I$ is nilpotent^{} (please, see parent object for details).

First of all, recall that $r(I)$ is an intersection^{} of all prime ideals^{} containing $I$ (please, see this entry (http://planetmath.org/ACharacterizationOfTheRadicalOfAnIdeal) for more details). Since $r(I)$ is maximal, it follows that there is exactly one prime ideal $P=r(I)$ such that $I\subseteq P$. In particular the ring $R/I$ has only one prime ideal (because there is one-to-one correspondence between prime ideals in $R/I$ and prime ideals in $R$ containing $I$). Thus, in $R/I$ an ideal $r(0)$ is prime.

Now assume that $\alpha \in R/I$ is a zero divisor. In particular $\alpha \ne 0+I$ and for some $\beta \ne 0+I\in R/I$ we have

$$\alpha \beta =0+I.$$ |

But $0+I\in r(0)$ and $r(0)$ is prime. This shows, that either $\alpha \in r(0)$ or $\beta \in r(0)$.

Obviously $\alpha \in r(0)$ (and $\beta \in r(0)$), because $r(0)$ is the only maximal ideal in $R/I$ (the ring $R/I$ is local). Therefore elements not belonging to $r(0)$ are invertible^{}, but $\alpha $ cannot be invertible, because it is a zero divisor.

On the other hand $r(0)=\{x+I\in R/I|{(x+I)}^{n}=0\text{for some}n\in \mathbb{N}\}$. Therefore $\alpha $ is nilpotent and this completes^{} the proof. $\mathrm{\square}$

Corollary. Let $p\in \mathbb{N}$ be a prime number and $n\in \mathbb{N}$. Then the ideal $({p}^{n})\subseteq \mathbb{Z}$ is primary.

Proof. Of course the ideal $(p)$ is maximal and we have

$$r\left(({p}^{n})\right)=r\left({(p)}^{n}\right)=(p),$$ |

since for any prime ideal $P$ (in arbitrary ring $R$) we have $r({P}^{n})=P$. The result follows from the proposition. $\mathrm{\square}$

Title | ideals with maximal radicals are primary |
---|---|

Canonical name | IdealsWithMaximalRadicalsArePrimary |

Date of creation | 2013-03-22 19:04:31 |

Last modified on | 2013-03-22 19:04:31 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Theorem |

Classification | msc 13C99 |