# zero ideal

The subset $\{0\}$ of a ring $R$ is the least two-sided ideal^{} of $R$. As a principal ideal^{}, it is often denoted by

$$(0)$$ |

and called the zero ideal^{}.

The zero ideal is the identity element^{} in the addition of ideals and the absorbing element in the multiplication of ideals (http://planetmath.org/ProductOfIdeals). The quotient ring^{} $R/(0)$ is trivially isomorphic^{} to $R$.

By the entry quotient ring modulo prime ideal, (0) is a prime ideal^{} if and only if $R$ in an integral domain^{}.

Title | zero ideal |

Canonical name | ZeroIdeal1 |

Date of creation | 2013-03-22 18:44:40 |

Last modified on | 2013-03-22 18:44:40 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 14K99 |

Classification | msc 16D25 |

Classification | msc 11N80 |

Classification | msc 13A15 |

Related topic | MinimalPrimeIdeal |

Related topic | PrimeRing |

Related topic | ZeroModule |