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Math Help - Nilradical in prime ideals

  1. #1
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    Nilradical in prime ideals

    Let R be a commutative ring, prove that each prime ideal of R contains the nilradical of R.

    Proof so far.

    Let I be a prime ideal of R, then if xy \in I , we would have either x or y in I.

    Now, nilradical of R is an element  r \in R such that r^n=0 \ \ \ n \geq 1 .

    How should approach this problem? I don't see how prime would lead to zero here. Thanks.
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  2. #2
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    Is 0 in I?
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  3. #3
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    Yes.

    So 0^1 = 0 .... Is that right? That sounds almost too easy...
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  4. #4
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    What you want to prove is that x \in Nil(R) \Rightarrow x \in I.
    But if x \in Nil(R), there is a n \in \mathbb{N} such that x^{n}=0
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