Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
4  
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
1  
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
4  
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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.