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This seems obvious to me, but I don't see it explicitly mentioned, so I may be wrong here, but since a prime ideal is primary, we have a primary decomposition for a prime ideal consisting of just the prime ideal and it's radical is itself. So this must be it's only associated prime, since the set of associated primes is independent of the particular primary decomposition.

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    @Georges Elencwajg: Perfect, thank you.2011-10-09

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