If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, then $QR_S$ is a radical ideal of $R_S$ If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, show $QR_S$ is a radical ideal of $R_S$.
Here is what I have done, since $Q$ is radical, let $q\in Q$ , then $q^n\in I$ for some $I$ ideal.
let $a\in QR_S$, then $a=q\cdot r/s$ for some $q\in Q$, $r\in R$ and $s\in S$.
then $a^n=q^n\cdot r^n/s^n\in IR_S$. since $IR_S$ is a ideal of $R_S$. we are done.
Am I right? and what I have confuse is why we need the condition of $Q∩S=∅$? I didn't use it at all. please help, thank you.
 A: $Q$ is a radical ideal if $\forall x\in R$ and $n\in \mathbb{N}$ such that $x^n\in Q$ it follows that $x\in Q$.
The condition for $Q\cap S=\emptyset$, assuming $S$ is a multiplicatively closed system has to be required for all ideals $P$ for which you construct $PR_S$ (as not to be equal to the whole ring) which can be proved to be an ideal of $R_S$. Otherwise, assume $\exists x\in Q\cap S\Rightarrow x\cdot 1/x\in QR_S\Rightarrow 1\in QR_S\Rightarrow QR_S=R_S$.
Now for the radical part, take $q\in R_S$, $q=a/s$, $a\in R$, $s\in S$ such that $\exists n\in\mathbb N$ such that $q^n\in QR_S\Rightarrow a^n/s^n=b/t, b\in Q, t\in S\Rightarrow  a^nt=bs^n\in Q\Rightarrow b^ns^n\in Q\Rightarrow a^nt\in Q\Rightarrow $
$a^nt^n\in Q\Rightarrow$ since $Q$ radical, $at\in Q\Rightarrow at/st=a/s\in QR_S$.
A: You can do this with or without resorting to elements. We outline how to do this with the structure of ideals.
We have $Q$ is radical iff $Q=\bigcap\{P\ |\ Q\subseteq P,\ P\text{ is a prime ideal in }R\}$.  An ideal in $R_S$ is prime iff is of the form $PR_S$ for prime ideal $P$ in $R$ such that $P\cap S=\emptyset$.  Since $Q\cap S=\emptyset$, the extended ideal $QR_S$ is proper and thus contained in a maximal ideal. Localization preserves the lattice structure of the ideals which are disjoint from $S$, hence $QR_S$ is the intersection of the primes in $R_S$ containing it, i.e., it is a radical ideal. 
I prefer this approach since it is conceptual rather than computational intensive. 
