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?

>>19
Is this what you're thinking of?

Let \{e_i\}_{i \in \mathbb{R}} be a basis and set a = f(e_0).  Write a/2 as a finite sum  \sum c_j e_j for c_j \in \mathbb{Z}.  Then a = 2(a/2) = \sum 2c_j e_j, which is different from e_0 since the coefficients are all even.

Yeah, that works.  Cool, thanks.