Odd. You titled this "field axioms" but Z is not a field!
Okay, from ac= bc, you get ac- bc= 0 and then you say
"ac-bc=c(a+(-b)) (I used the axiom associative)"
Well, first, that's the distributive axiom, not associative. But also you don't say that is equal to 0 which you should.
Now, which axiom, specifically, allows you to say that "if c(a- b)= 0 and c is not 0 then a- b= 0"?