Quotient Field as localization

1. Proposition

Let 20230724-quotient_field_as_localization_6130b692b4d58a44d70e323b9432701598ebebee.svg be an integral domain and 20230724-quotient_field_as_localization_572a154b9e6adca718f2b5f4af24c325b4be8d26.svg be a commutative ring TFAE:

  1. 20230724-quotient_field_as_localization_122bd3beaed246f4b39676e6fd290e9124cf72fd.svg is the quotient field / Quotient Fied (?)
  2. 20230724-quotient_field_as_localization_762922a759ebd3849cffe8bcb19374b9d345d036.svg is the localization along 20230724-quotient_field_as_localization_5ba19af37a6003ce5dbf5d8e858a81a32850d7e5.svg

2. Proof

similar construction, aside from multiplicatino with 20230724-quotient_field_as_localization_17f0e938177590d1af9f393da369dd91795a8749.svg. But 20230724-quotient_field_as_localization_ac8714a8e572f776592e141bede62cfe72a80cda.svg is an integral domain, hence 20230724-quotient_field_as_localization_0649d79c18d47bcea080e00a37109e99a17aca3e.svg

alternatively

20230724-quotient_field_as_localization_fb03577c1eb6a4220673df38e364fc4e3eb3188e.svg

Date: nil

Author: Anton Zakrewski

Created: 2024-10-13 So 15:22