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Thread: prime ideal

  1. #1
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    prime ideal

    Let $\displaystyle S$ be a graded ring.
    $\displaystyle f$ : homogeneous element of degree $\displaystyle > 0$.
    $\displaystyle S_f$= localized ring $\displaystyle T^{-1}S$, where $\displaystyle T$={$\displaystyle f^k|k \ge 0$}.
    $\displaystyle S_{(f)}$ = {elements of degree zero in $\displaystyle S_f$}.
    Then there is 1-to-1 corresponence between homogeneous prime ideals of $\displaystyle S$ which does not contain $\displaystyle f$ and pime ideals of $\displaystyle S_{(f)}$.

    In my opinion, homogeneous prime ideal $\displaystyle P$ in $\displaystyle S$ may relate to pime ideal $\displaystyle (P S_f)\cap S_{(f)}$of $\displaystyle S_{(f)}$.
    But I can't show that any prime ideal of $\displaystyle S_{(f)}$ is of the form $\displaystyle (P S_f)\cap S_{(f)}$.
    Last edited by Stiger; Feb 5th 2010 at 06:28 AM.
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