Is it true that a ring has no zero divisors iff the right and left cancellation laws hold?

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.

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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.

Suppose $ab = 0$ with $a, b \ne 0$. Either $ba = 0$ (which means $a$ and $b$ are zero-divisors), or $ba \ne 0$, in which case $ba$ is a zero-divisor because $a(ba) = 0$ and $(ba)b = 0$.