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Math Help - localization of a ring

  1. #1
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    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}
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  2. #2
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    Quote Originally Posted by student2011 View Post
    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}.
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