I'm an undergrad mathematics student and I'd like to request some books about Sylvester domains. Specifically I'd like to understand the fact that not all modules are Sylvester domains. I just proved Sylvester's nullity theorem and I didn't use any reference to the reals. Here's the proof:
Let A and B be square matrices. Prove: $$\text{rank} (AB) \geq \text{rank} (A)+ \text{rank}(B)-n$$
It is trivial to prove: $$\text{rank} A \geq \text{rank}(AB)$$ The desired result then follows.
Two things:
"Not all modules are Sylvester domains" Well, modules aren't usually rings at all. It would make more sense to understand "not all domains are Sylvester domains."
Secondly, the description of Sylvester's law of nullity talks about rectangular matrices, not just square matrices, so that throws some doubt on the above idea.
You can definitely find a proof more general than just the real numbers because any field will work, as shown by Sylvester. You can go even further than fields. After you leave fields behind, you need to be more careful with your intuition about how facts about matrices will work.
One book that has extensive information on Sylvester domains is Cohn's Free Ideal Rings book starting page 291. It gives Weyl algebras as examples of domains that aren't Sylvester domains.