# Math Help - localization of a ring

1. ## localization of a ring

Show that if $S$ is any multiplicative closed subset of $R$ where $R$ is commutative ring with unity, $(\sqrt{I})_s=\sqrt{I_s}$

2. Originally Posted by student2011
Show that if $S$ is any multiplicative closed subset of $R$ where $R$ is commutative ring with unity, $(\sqrt{I})_s=\sqrt{I_s}$
if $a \in \sqrt{I}$, then $a^n \in I$ for some integer $n$. so if $s \in S$, then $(s^{-1}a)^n=s^{-n}a^n \in S^{-1}I$. thus $s^{-1}a \in \sqrt{S^{-1}I}$.
this proves that $S^{-1} \sqrt{I} \subseteq \sqrt{S^{-1}I}$.
for the converse, let $s \in S$ and $a \in R$ be such that $s^{-1}a \in \sqrt{S^{-1}I}$. then $s^{-n}a^n \in S^{-1}I$, for some integer $n$. so $s^{-n}a^n = t^{-1}b$, for some $t \in S$ and $b \in I$. thus $uta^n = us^n b \in I$, for some $u \in S$. thus $(uta)^n \in I$ and hence $uta \in \sqrt{I}$.
therefore $s^{-1}a=(sut)^{-1}uta \in S^{-1} \sqrt{I}.$