Let $R$ be a ring with unity and assume that $R$ has no nonzero zero-divisors. Let $a,b\in R$, and assume that $ab=1$. Show that $ba=1$, and therefore $a,b$ are units.
I think this question boils down to showing that $R$ is communitive under multiplication (of the ring R), but I don't know how to show it given the conditions. Can someone help please? Thanks