# integral

Let $B$ be a ring with a subring $A$. We will assume that $A$ is contained in the center of $B$ (in particular, $A$ is commutative^{}). An element $x\in B$ is *integral* over $A$ if there exist elements ${a}_{0},\mathrm{\dots},{a}_{n-1}\in A$ such that

$${x}^{n}+{a}_{n-1}{x}^{n-1}+\mathrm{\cdots}+{a}_{1}x+{a}_{0}=0.$$ |

The ring $B$ is *integral* over $A$ if every element of $B$ is integral over $A$.

Title | integral |
---|---|

Canonical name | Integral |

Date of creation | 2013-03-22 12:07:44 |

Last modified on | 2013-03-22 12:07:44 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 9 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 13B21 |

Related topic | IntegralBasis |