So we let R be a commutative ring with identity and let S = 1 + I be a multiplicatively closed set. I'm trying to establish a homeomorphism between m-spec(S^{-1}R) and m-spec(R/I). It seems relatively clear that there is a bijection between these sets, but I'm struggling with defining such a function so that I can show the homeomorphism.
So it seems that they both share a bijection with V(I) intersect with m-spec(R), where V(I) denotes all the prime ideals containing the ideal I. I think I can construct a homeomorphism from this intersection to m-spec(R/I), but the struggle is really showin…
