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?

>>12
>That fact that Z is not divisible doesn't imply it's not ring isomorphic to the rationals.

Divisibility, (or equivalently in this case, the property of being a field), is preserved under isomorphism.

>R has multiplicative elements of finite order, take -1 for example.

Torsion subgroup has infinite order, again something that's preserved under isomorphism.

>>11
What I want to say is that R and C are both uncountably-infinite dimensional free Z-modules and therefore isomorphic since they have the same dimension, but I'm not actually sure R is free over Z...

Hmmm.