If a=b, how does a-b=0?

Question

Let R be a ring and let $a,b \in R$.

Suppose $a=b$. How do you know that $a +- b=0$?

In other words, what axiom or step lets you say that $a +-b = b + -b$? (That you can add the same thing to both sides of the equation.)

Answer 1

Making substitutions like that is always allowable. Essentially, all we're trying to say there is that $$a=b$$ implies $$f(a)=f(b)$$ (where $f$ here is "add $-b$"). But we don't need any special axioms for our ring to do this; this is just how equality works in all cases.