Localization at a prime ideal

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Definition

Definition with symbols

Let A be a commutative unital ring and P a prime ideal in A. Then, the localization of A at P is defined as follows:

  • As a set, it is the collection of fractions a/s where a \in A, s \in A \setminus P, subject to the equivalence a/s \sim a'/s' \iff as' = a's.
  • The operations are defined as follows: a/s + a'/s' = (as' + a's)/(ss') and (a/s) (a'/s') = (aa')/(ss')

Facts

A embeds naturally as a subset of A_P. If P \le P' are prime ideals in A, we have an embedding from A_{P'} into A_P. In fact, if A is an [[integral domain], then all the A_Ps are contained inside the residue field of A. Further, their intersection is exactly A.