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 $a/s\sim a'/s'\iff as'=a's$.
• 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 $P\leq P'$ 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 $A$.