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???
Hi, thank you very much for helping!
I get that f is a well-defined homomorphism and that it is surjective, only... it seems to me that f sends every element of the form
(s,t)^(-1)(0,b) to s^(-1)*0 = 0 Am I doing something wrong here?