Artin states "Let $R$ be a ring, $f$ be a monic polynomial and let $g$ be any polynomial, both with coefficients in $R$. There are uniquely determined polynomials $q$, $r$ in $R[x]$ such that $g = fq + r$ with degree $r <$ degree $f$ or $r = 0$."
He does not prove this theorem. The part I am having trouble with is uniqueness.
For the proof in a field, you assume there exists $q_1$, $r_1$ and $q_2$, $r_2$ and show that $$(q_1 - q_2)f = r_1 - r_2$$ In a field this proves that $q_1 = q_2$ and $r_1 = r_2$ due to degree LHS > degree RHS unless both zero. However, in an arbitrary ring I am not certain due to zero divisors.
Thanks for the help.