An integral domain has sub domains. All together you get ten. We know there is an close connection between groups and manifolds. Can you have a kind of Mayer-Vietoris sequence between domains? What I mean is let's say you have a map from an integral domain to a commutative ring, and another map to another commutative ring. And then you have a map from either commutative ring into a larger group. So now can you put an "H" (homology groups) in front of each map? And if so, would this work for all the sub-rings underneath the integral domain? For example could you do the same with a Noetherian Ring, and another Noetherian Ring. And you have an intersection between them. And you map into some larger ring. Can you apply the Mayer Vietoris theorem? Or would there be an obstruction?