# nilradical

Let $R$ be a commutative ring. An element $x\in R$ is said to be nilpotent if ${x}^{n}=0$ for some positive integer $n$. The set of all nilpotent elements of $R$ is an ideal of $R$, called the nilradical of $R$ and denoted $\mathrm{Nil}(R)$. The nilradical is so named because it is the radical^{} of the zero ideal^{}.

The nilradical of $R$ equals the prime radical of $R$, although proving that the two are equivalent^{} requires the axiom of choice^{}.

Title | nilradical |
---|---|

Canonical name | Nilradical |

Date of creation | 2013-03-22 12:47:52 |

Last modified on | 2013-03-22 12:47:52 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 4 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 13A10 |

Related topic | PrimeRadical |

Related topic | JacobsonRadical |

Defines | nilpotent |