f: Spec(S) -> Spec(R) How do I prove the homomorphism between the two prime spectrums of R and S is continuous? I have a strategy, but I'm having problems trying to see how to formulate a proof. My strategy is as follows. I was able to prove earlier in the assignment that a prime ideal P in S will get mapped back to a prime ideal f-1(P) in R. From this I gather that there will be some ideal A contained in a collection of prime ideals V(B) both in R. Then f-1(A)=f-1(V(A)) and that's going to be in P which is in S. I think. As of right now this is my "gut reaction" to seeing the question. **V(I) is the collection of all prime ideals of R that contain I.