I am unsure what your first question is asking. I mean, an isomorphism is entirely disjoint from the sets A and B which your rings are defined over. For example, one can take the ring generated by 3 in , and the ring generated by 2. These have different orders, and so are non-isomorphic. On the other hand, one can take the ring and it is isomorphic to the ring consisting of 2-by-2 matrices,
where . The sets you are working in don't matter!
I think I would recommend you looking up the computer algebra package GAP. It often tells you how it does stuff in the manual.