# Localization at a prime ideal

From Commalg

## Definition

### Definition with symbols

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

- As a set, it is the collection of fractions where , subject to the equivalence .
- The operations are defined as follows: and

## Facts

embeds naturally as a subset of . If are prime ideals in , we have an embedding from into . In fact, if is an [[integral domain], then all the s are contained inside the residue field of . Further, their intersection is exactly .