Theorem 19.5

Statement

The cancellation laws hold in a ring $R$ if and only if $R$ has no divisors of $0$.

Proof

Let $R$ be a ring in which the cancellation laws hold, and suppose $ab=0$ for some $a,b \in R$. We must show that either $a$ or $b$ is $0$. If $a\neq 0$, then $ab=a0$ implies that $b=0$ by cancellation laws. Similarly, $b\neq 0$ implies that $a=0$, so there can be no divisors of $0$ if the cancellation laws hold.

Conversely, suppose that $R$ has no divisors $0$, and suppose that $ab=ac$ with $a\neq 0$. Then

(1)
$$ab-ac=a(b-c)=0.$$

Since $a \neq 0$, and since $R$ has no divisors of $0$, we must have $b-c=0$, so $b=c$. A similar argument shows that $ba=ca$ with $a \neq 0$ implies $b=c$.