i'm working through the following text and I think I found an error please let me know if i'm totally wrong.(adsbygoogle = window.adsbygoogle || []).push({});

Janusz, Gerald J. Algebraic Number Fields

and i'm starting with the 3rd exercise on page 3. It is as following:

let R be an integral domain and p a prime ideal of R. Show there is an isomorphism between the fields R/p and R_p/(pR_p).

Note that R_p is the localization at the multiplicative set S=R-p (this is multiplicative b/c p is a prime ideal). Now I believe that pR_p is a maximal ideal: this follows because if it were strictly contained in a maximal ideal of R_p then this would be a prime ideal of R_p different from pR_p but this corresponds to a prime ideal of R distinct from p. This corresponding prime ideal in R is in our multiplicative set at which we localized so we have invertible elements in a maximal ideal, a contradiction (maximality implies proper containment).

I used the fact that there is a bijection between prime ideals of R_p and prime ideals of R that don't intersect S. Also that all maximal ideals are prime.

So I believe that R_p/pR_p is a field (because it's a quotient of a ring by a maximal ideal). But there is nothing to suggest that p is a maximal ideal (in a general ring with a general ideal a quotient R/p is a field if and only if p is maximal) unless we are in a PID (because then prime ideals are maximal). So if this result were true it would mean that in ANY integral domain a prime ideal is maximal i.e. that the quotient of an integral domain by one of it's prime ideals is a field, there is an easy counterexample.

Z[x] is an integral domain and <x> is prime (because Z[x]/<x>=Z is an integral domain) but Z[x]/<x>=Z is most definitely NOT a field.

What do you think?

Cheers

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# Localization in integral domains

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