# nil and nilpotent ideals

An element $x$ of a ring is *nilpotent ^{}* if ${x}^{n}=0$ for some positive integer $n$.

A ring $R$ is *nil* if every element in $R$ is nilpotent. Similarly, a one- or two-sided ideal^{} is called *nil* if each of its elements is nilpotent.

A ring $R$ [resp. a one- or two sided ideal $A$] is *nilpotent* if ${R}^{n}=0$ [resp. ${A}^{n}=0$] for some positive integer $n$.

A ring or an ideal is *locally nilpotent* if every finitely generated^{} subring is nilpotent.

The following implications^{} hold for rings (or ideals):

$$\text{nilpotent}\mathit{\hspace{1em}}\Rightarrow \text{locally nilpotent}\mathit{\hspace{1em}}\Rightarrow \text{nil}$$ |

Title | nil and nilpotent ideals |

Canonical name | NilAndNilpotentIdeals |

Date of creation | 2013-03-22 13:13:25 |

Last modified on | 2013-03-22 13:13:25 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 16N40 |

Related topic | KoetheConjecture |

Defines | nil |

Defines | nil ring |

Defines | nil ideal |

Defines | nil right ideal |

Defines | nil left ideal |

Defines | nil subring |

Defines | nilpotent |

Defines | nilpotent element |

Defines | nilpotent ring |

Defines | nilpotent ideal |

Defines | nilpotent right ideal |

Defines | nilpotent left ideal |

Defines | nilpotent subring |

Defines | locally nilpotent |

Defines | locally nilpotent ring |

Defines | locally nilpo |