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Thread: properties of radical of an ideal

  1. #1
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    properties of radical of an ideal

    Hello everybody;

    Let $\displaystyle R$ be commutative ring with unity and let $\displaystyle I$ and $\displaystyle J$ be two ideals of $\displaystyle R$ Show that

    $\displaystyle \sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$
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  2. #2
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    Quote Originally Posted by student2011 View Post
    Hello everybody;

    Let $\displaystyle R$ be commutative ring with unity and let $\displaystyle I$ and $\displaystyle J$ be two ideals of $\displaystyle R$ Show that

    $\displaystyle \sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$
    $\displaystyle \sqrt{I+J} \subseteq \sqrt{\sqrt{I}+\sqrt{J}}$ is obvious. for the other inclusion, let $\displaystyle a \in \sqrt{\sqrt{I}+\sqrt{J}}$. then there exists a positive integer $\displaystyle n$ such that $\displaystyle a^n = b+ c$, for some $\displaystyle b \in \sqrt{I}$ and $\displaystyle c \in \sqrt{J}$. so $\displaystyle b^r \in I$ and $\displaystyle c^s \in J$, for some positive integers $\displaystyle r, s$. now show that $\displaystyle a^{n(r+s)} \in I + J$ and thus $\displaystyle a \in \sqrt{I+J}$.
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  3. #3
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    Thank you very much, I catch the point. In order to show that $\displaystyle a^{n(r+s)}\in I+J$, we do the following:

    $\displaystyle (b+c)^{r+s}=\sum_{k=0}^{r+s}\binom{r+s}{k}\times b^{r+s-k}\times c^{k} \in I+J$


    This shows that $\displaystyle a^{n(r+s)} \in I+J$
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