# characterisation

In mathematics, characterisation usually means a property or a condition to define a certain notion. A notion may, under some presumptions, have different ways to define it.

For example, let $R$ be a commutative ring with non-zero unity (the presumption). Then the following are equivalent^{}:

(1) All finitely generated^{} regular ideals of $R$ are invertible^{}.

(2) The $(a,b)(c,d)=(ac,bd,(a+b)(c+d))$ for multiplying ideals of $R$ is valid always when at least one of the elements $a$, $b$, $c$, $d$ of $R$ is not zero-divisor.

(3) Every overring of $R$ is integrally closed^{}.

Each of these conditions is sufficient (and necessary) for characterising and defining the Prüfer ring.

Title | characterisation |
---|---|

Canonical name | Characterisation |

Date of creation | 2013-03-22 14:22:28 |

Last modified on | 2013-03-22 14:22:28 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 18 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 00A05 |

Synonym | characterization |

Synonym | defining property |

Related topic | AlternativeDefinitionOfGroup |

Related topic | EquivalentFormulationsForContinuity |

Related topic | MultiplicationRuleGivesInverseIdeal |