Originally Posted by

**Joolz** Hi there,

I would like some help with the following:

(note: all rings are commutative)

R is defined to be the direct product of A and B. So R= A x B

Let p be a prime ideal of the form p=q x B where q is a prime ideal in A.

Then the fraction rings Rp and Aq are isomorphic. Intuitively I can see that it is indeed true, but I get stuck at the proof.

I tried to find homomorphisms f: R ---> Rp, g: R ---> Aq for which there exist a unique homomorphism h: Rp --->Aq with hf=g

I tried to let f be the localization map which sends (a, b) to (a/1, b/1) and g sends (a,b) to (a/1) but then get an h that is very much not surjective...

So, first of al: what do the elements of Rp look like? I thought (a/s, b) where s in A\q but I'm not sure...

Or do I need to use the fact that Rp and Aq are local rings???