# product of ideals

Let $R$ be a ring, and let $A$ and $B$ be left (right) ideals of $R$. Then the product of the ideals $A$ and $B$, which we denote $AB$, is the left (right) ideal generated by^{} all products $ab$ with $a\in A$ and $b\in B$. Note that since sums of products of the form $ab$ with $a\in A$ and $b\in B$ are contained simultaneously in both $A$ and $B$, we have $AB\subset A\cap B$.

Title | product of ideals |

Canonical name | ProductOfIdeals |

Date of creation | 2013-03-22 11:50:59 |

Last modified on | 2013-03-22 11:50:59 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 11 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 16D25 |

Classification | msc 15A15 |

Classification | msc 46L87 |

Classification | msc 55U40 |

Classification | msc 55U35 |

Classification | msc 81R10 |

Classification | msc 46L05 |

Classification | msc 22A22 |

Classification | msc 81R50 |

Classification | msc 18B40 |

Related topic | SumOfIdeals |

Related topic | QuotientOfIdeals |

Related topic | PruferRing |

Related topic | ProductOfLeftAndRightIdeal |

Related topic | WellDefinednessOfProductOfFinitelyGeneratedIdeals |