Cancellation laws in Rings

Suppose that $ab = ac$ implies $b = c$ for non-zero $a$. Then $a(b-c) = 0$ implies $b-c = 0$. Suppose $a$ is a left zero divisor, i.e. $ad = 0$ for some non-zero $d$. Then taking $b = c+d$ gives $a(b-c) = ad = 0$. So $c = b = c+d$, so $d = 0$, which is a contradiction. So left cancellation implies that the ring has no zero divisors.

Similiarly, right cancellation also implies that the ring has no zero divisors.

Conversely, if a ring has no zero divisors (i.e. it is a domain) then $ab = ac$ ($a \neq 0$) implies $a(b-c) = 0$ implies $b - c = 0$ as $a$ is non-zero and there are no zero divisors, hence $b - c$. Similiarly right cancellation holds.

Conclusion: the following are equivalent for a ring $R$: $R$ has no zero divisors; the left cancellation law holds in $R$; the right cancellation law holds in $R$.