Localization at a prime ideal

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 $$A_P$$s are contained inside the residue field of $$A$$. Further, their intersection is exactly $$A$$.