I don't think you have shown what is required. To show that this ring is an integral domain, we must show that it has no non-zero divisors, i.e.
So, we assume that and that . But then . This is a contradiction, so either or must be zero.
Your definition of cancellation is a bit badly formed. You said
" if a, b, and c are elements of R, then a does not equal to 0 and ab = ac always imply b = c "
This is wrong: the fact that is not a consequence of the fact that , it is part of the definition of cancellation. Compare to the following:
" if and then ab=ac always implies b = c "
Do you see the difference?