This is the definition of zero divisor in Hungerford's Algebra:
A zero divisor is an element of $R$ which is BOTH a left and a right zero divisor.
It follows a statement:
It is easy to verify that a ring $R$ has no zero divisors
if and only if the right and left cancellation laws hold in $R$;
that is,
for all $a,b,c\in R$ with $a\neq 0$,
$$ab=ac~~~\text{ or }~~~ba=ca~~~\Rightarrow~~~ b=c.$$
I think it is not true.
But I can't find a counterexample.
Best Answer
Lemma: A ring has a left (or right) zero-divisor if and only if it has a zero divisor.
Proof: Assume $ab=0$ for $a,b\neq 0$.
If $ba=0$, you are done - $a$ is both a left and right zero divisor.
If $ba\neq 0$, then $a(ba)=(ab)a=0$ and $(ba)b=b(ab)=0$, so $ba$ is a left and right zero divisor.
Now it is much easier to prove your theorem.
If $ax=ay$ and $R$ has no zero-divisors, then $a(x-y)=0$. But, by the lemma, $R$ also has no left-zero divisors, so either $a=0$ or $x-y=0$.
Similarly for $xa=ya$.
On the other hand, if cancellation is true, then $a\cdot b=0=a\cdot 0$ means that either $a=0$ or $b=0$. So there can't be any left zero divisors, and thus no zero divisors.