Localization at a prime ideal

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 $a/s\sim a^{\prime }/s^{\prime }⟺a{s}^{\prime }=a^{\prime }s$.
• The operations are defined as follows: $a/s+a^{\prime }/s^{\prime }=(a{s}^{\prime }+a^{\prime }s)/(s{s}^{\prime })$ and $(a/s)(a^{\prime }/s^{\prime })=(a{a}^{\prime })/(s{s}^{\prime })$

Facts

$A$ embeds naturally as a subset of $A_P$. If $P\le P^{\prime }$ are prime ideals in $A$, we have an embedding from $A_{P^{\prime }}$ 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 $A$.