Cancellation laws in Rings

Question

In rings left and right cancellation laws generally don't hold.

can anyone generalize some cases so that we are ensured when the cancellation laws hold in rings?(the case I found was in Integral domains they hold.)

Answer 1

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 $R$.

Answer 2

Let $R $ be a ring with cancellation laws holding.

Let $a,b\in R$.

Now,if $ab=ac$ then by left cancellation law we get,$b=c$.

Similarly,if $ba=ca$ then by right cancellation law we get,$b=c$.

Let $ab=0$ then if $a\neq 0$.So,$ab=a0,$by left cancellation law $b=0$.

Therefore,if $ab=0$ then either $a=0$ or $b=0$.