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Math Help - properties of radical ring

  1. #1
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    properties of radical ring

    Show that \sqrt{I}=\cap{P}, where the intersection is over all prime ideals of R that contain I.

    I know how to verify the one inclution \sqrt{I}\subset\cap{p}, but how can I show the second inclution.


    Thank you in advance
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint:

    The nilradical of R is the intersection of all prime ideals of R . Apply this result to R/I .
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  3. #3
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    Thaaaaank you very much, only now I get the point.

    L(R/I)=\cap{(P/I)} where P is prime ideal in  R contains I. But we know that \cap{(P/I)}=\cap{P}/I . Thus, if x \in \cap{P}, then x+I \in \cap{P} +I=L(R/I). It is easy to show that L(R/I)=\sqrt{I}.


    Thank you again
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