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If K is an algebrically closed field, Is any prime ideal of K[X1,...,Xn] the intersection of a finite number of maximal ideals ?1

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No, take the zero ideal. But even in non-pathological cases, you ask if every irreducible affine variety is a finite set, which is of course wrong in general. – Martin Brandenburg May 31 '11 at 6:49
Isn't it true that if $n>1$ this is never the case for non-maximal primes? – the L May 31 '11 at 6:52
Liran -- it is. – algori May 31 '11 at 6:53
You could parse the question differently, and then the answer is "Yes, a maximal ideal is an example of a prime ideal that is the intersection of finitely many maximal ideals." – S. Carnahan May 31 '11 at 7:01
In any $T_0$ topological space the only finite irreducible sets are the singletons, hence the only primes satisfying what you want correspond to points i.e., maximal ideals. – Guillermo Mantilla May 31 '11 at 8:58

In any commutative ring, any ideal that is the intersection of two other ideals is not prime. Therefore, a prime ideal is the intersection of finitely many maximal ideals if and only if it is itself maximal.

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