 Multiplicative cancellation in an integral domain

In an integral domain, $ab=ac$ with $a\ne 0$ gives $a(b-c)=0$ by the distributive property. Since there are no zero divisors, $b-c=0$. Conversely, in a nonzero commutative ring, this cancellation property rules out zero divisors by comparing $ab=0$ with $a0=0$.