I'd like to know something...

How can I show that the Zero product property implies or holds in the Integers?

I'd like to show that so I can show that the rationals are a field (without getting into the real numbers)...but all the proofs I've seen about the Zero product property are using field axioms. I don't know if I can show that (The only thing that seems to fail in the proof is the closure of the sum given the possibility that the denominator after the sum becomes zero if I don't prove that it doesn't happens in the integers).

So I don't know how to do it in a ring.