So I'm kind of confused about the definition:
[tex] a-b\in I[/tex]
Why a - b instead of a + b?
Are you trying to prove that a-b is an equivalence relation? Can you be more specific?
It's actually natural to write a-b because I becomes zero when you quotient out by it, so a-b in I means a-b=0 or, written differently, a=b.
you would presumably want a equivalent to a, which follows from a-a in I, rather than a+a in I.
Separate names with a comma.