abstract algebra question

Is there a shortcut to prove that Z, Q, R, S are pairwise nonisomorphic? I can split it into three proofs and do them individually, but I feel like there has to be some property that covers all the cases.

It can't be just the fact that Z < Q < R < S because the subset of even integers is isomorphic to the set of all integers.

Anyone care to push me in the right direction, if there is one?

>>18
Im almost sure that R or C arent free over Z. Supose they are; let f be an iso between direct sums of Z and R. take an element e from the basis of sums of Z. Then f(e) = a, for a=!0. take f^-1 (a/2). Then a = f(e) and a = 2*f(f^-1(a/2)), making a contradiction.