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Math Help - Radical

  1. #1
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    Radical

    Let F be a field and let I and J be ideals in a ring F[x_1,...,x_n].
    Generally, \sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I}+\sqrt{J}.

    Give an example in F[x_1,...,x_n] where \sqrt{I+J} \ne \sqrt{I}+\sqrt{J}.
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  2. #2
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    Quote Originally Posted by KaKa View Post

    Let F be a field and let I and J be ideals in a ring F[x_1,...,x_n].
    Generally, \sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I}+\sqrt{J}. false!
    who told you that? for example in \mathbb{R}[x,y] let I=<x^2+y^2>, \ J=<y>. then both I,J are prime ideals and hence \sqrt{I}=I, \sqrt{J}=J and so \sqrt{I}+\sqrt{J}=I+J=<x^2,y>.

    but \sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I+J}=\sqrt{<x^2,y>}  =<x,y> \neq <x^2,y>.


    Give an example in F[x_1,...,x_n] where \sqrt{I+J} \ne \sqrt{I}+\sqrt{J}.
    same as above.
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