# extended ideal

Let $f:A\to B$ be a ring map. We can look at the ideal generated by the image of $\U0001d51e$, which is called an extended ideal and is denoted by ${\U0001d51e}^{e}$.

It is not true in general that if $\U0001d51e$ is an ideal in $A$, the image of $\U0001d51e$ under $f$ will be an ideal in $B$. (For example, consider the embedding $f:\mathbb{Z}\to \mathbb{Q}$. The image of the ideal $(2)\subset \mathbb{Z}$ is not an ideal in $\mathbb{Q}$, since the only ideals in $\mathbb{Q}$ are $\{0\}$ and all of $\mathbb{Q}$.)

Title | extended ideal |
---|---|

Canonical name | ExtendedIdeal |

Date of creation | 2013-03-22 12:55:34 |

Last modified on | 2013-03-22 12:55:34 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 6 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 13A15 |

Classification | msc 14K99 |

Related topic | ContractedIdeal |