# complete ring of quotients of reduced commutative rings

There is a characterization of complete ring of quotients of reduced commutative rings. Let $A$ be a reduced (http://planetmath.org/ReducedRing) commutative ring, then if $B$ is an overring of $A$ and if for any element $b\in B\backslash \{0\}$ there is an $a\in A$ such that $ab\in A\backslash \{0\}$, then $B$ is said to be a *rational extension* of $A$. See how similar this is with the definition of essential extension^{} in the category of rings, obviously all rational extensions of reduced commutative rings are also essential extensions. Furthermore there is a maximum (upto $A$-isomorphism^{}) rational extension of $A$ and this is in fact the complete ring of quotients of $A$.

Title | complete ring of quotients of reduced commutative rings |
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Canonical name | CompleteRingOfQuotientsOfReducedCommutativeRings |

Date of creation | 2013-03-22 18:27:33 |

Last modified on | 2013-03-22 18:27:33 |

Owner | jocaps (12118) |

Last modified by | jocaps (12118) |

Numerical id | 6 |

Author | jocaps (12118) |

Entry type | Theorem |

Classification | msc 13B30 |

Related topic | CompleteRingOfQuotients |

Related topic | essentialmonomorphism |

Defines | rational extension |