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?

>>25
A Z-module and an abelian group are the same thing.

R and C have the same dimension as Q-vector spaces, and therefore are isomorphic as Q-vector spaces (but not as rings).  A Q-vector space isomorphism is also a Z-module isomorphism, i.e. an abelian group isomorphism.