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Prime ideals of tensor products
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cliomseerg@kriocoucke.mai
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 128

PostPosted: Tue May 30, 2006 9:18 am    Post subject: Prime ideals of tensor products Reply with quote

Dear NG,

If I have commutative rings and I know their prime ideals, and I know
that I can make a tensor product between them with respect to some ring
(say Z), then can I completely determine the prime ideals in this
tensor product?

Sincerely,
Jose Capco
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Jannick Asmus
science forum Guru


Joined: 25 Mar 2005
Posts: 312

PostPosted: Mon Jun 05, 2006 9:48 pm    Post subject: Re: Prime ideals of tensor products Reply with quote

On 30.05.2006 11:18, Jose Capco wrote:
Quote:
Dear NG,

If I have commutative rings and I know their prime ideals, and I know
that I can make a tensor product between them with respect to some ring
(say Z), then can I completely determine the prime ideals in this
tensor product?

I think this is not possible in general. E.g., let R=Z[X] and K be any
field, we have K(x)_Z Z[X] = K[X] and the non-trivial prime ideals in
K[X] are generated by irreducible polynomials which cannot be coded by
any prime ideals in Z[X] and K.

Nevertheless, if R and S are finitely generated algebras over a field k,
then there is a link between the maximal ideals of the tensor product
R(x)S (over k) and the set theoretic product of the maximal spectra of R
and S. This is one of the starting points of the theory of varieties.

Quote:

Sincerely,
Jose Capco


HTH.

Best wishes,
J.
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