> [!NOTE] Lemma (Cancellation Law for Integral Domains) > Let $R$ be an [[Integral Domain|integral domain]]. For all $a,b,c\in R$ with $a \neq 0_{R}$, if $ab=ac$ then $b=c$. ###### Proof We have $ab-ac=0$. So $a(b-c)=0$. We have $b-c=0_{R}$ since otherwise $b-c$ is a zero divisor giving a contradiction. Thus $b=c$.