Cancellation in integral domains

Cancellation in integral domains: Let $D$ be an integral domain and let $a,b,c\in D$ with $a\neq 0$. If $ab=ac$, then $b=c$. Likewise, if $ba=ca$, then $b=c$.

In an integral domain , every nonzero element is a regular element (equivalently, there are no nonzero zero divisors ), and this is exactly what makes cancellation possible. The same injectivity argument is the starting point for constructing the field of fractions .