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I am not that well acquainted with commutative algebra, so that I am kind of struggling to understand the concept of "localizing a ring at a prime ideal."

For one thing I can say is that localizing alludes to a geometric meaning.

  1. Can somebody clarify this concept from both perspectives: algebraic and geometric ?

  2. Since a ring ( for example $\mathbb{Z}$ ) can contain many prime ideals, I presume that the introduced localization is not unique. Is this correct ? What are the consequences ? How then do you make a choice on picking up the most suitable prime ideal for the purpose of the localization ?

  3. How do you procede in constructing such "localization" ?

Many thanks.

user249018
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  • Also this post and this post. Please try the search feature first next time. – rschwieb Feb 08 '18 at 21:50
  • As for 2) Yes, there is no "the localization of a ring" it is "the localization of the ring with respect to a prime ideal" Actually, you can just use multiplicative sets, but complements of prime ideals are the most informative such sets. – rschwieb Feb 08 '18 at 21:51
  • As for 3), in what context are you learning about localizations in which the localization isn't constructed for you? Anyway, the answer to your question 3) is in every commutative algebra book, and extensively described at the wiki – rschwieb Feb 08 '18 at 21:52

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