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?

>>26
>>27
Well, thanks a lot, and also, your moms.

Seriously speaking now, I knew for the Z-module-abelian group equivalence.  I think the main problem is my lack of knowledge of infinite cardinals and my lack of knowledge in module things. If I remember my notes, module are like a generalisation of vector space amirite ?

Ah btw, since you guys know quite a few things in algebra, I would like to ask you if what is the difference beteween a "unitary ring" and between an "algebra", aside from teh names. I mean, if I look at the definitions, ring needs two associative laws, one with a commutative group property and the other with just a group property, and is then stable for its two laws. And the definition of an algebra is the same : stability for a sum law, stability for a product law, for linear combination of elements of the algebra, but it just seems that the definition of an algebra include the action of a vector space on the algebra...