Let $R$ be a ring with unity and $a, b \in R$. Assume $a$ and $b$ are not zero divisors. Show that $a$ and $b$ are units, if $ab$ is a unit.
Clearly, $a$ has a right inverse etc., but $R$ is not required to be commutative. Evidently, it's necessary to use the condition that the elements are not zero divisors, but I'm not getting anywhere.