Question about finite ring with more than one element; division ring 
A finite ring with more than one element and no zero divisors is a division ring. (Hungerford, Algebra, Exercise 6, Chapter 3, Section 1.)

This a problem taken from Hungerford's graduate algebra text. Hungerford defines left and right zero divisors and then says that an element is a zero divisor if it is both a left and right zero divisor. I took a peek at a solution to the exercise and the author of the solution says that for $r \in R-\{0\}=R^*$, $rR^*=R^*=R^*r$. I might be missing something. 
Say I want to show $rR^*=R^*$. Pick $ry \in rR^*$. If $ry=0$ then this implies that $r$ is a left zero divisor and $y$ is a right zero divisor. 

$R$ has no zero divisors, but I am assuming this doesn't exclude the possibility of elements being a right zero divisor or a left zero divisor. I am thinking correctly here? 

Just answering this step is fine, I think I can try to continue working on the problem from this point. Thanks.


Lemma: A ring with no nonzero two-sided zero divisors has no nonzero one-sided zero divisors.

Suppose $ab=0$ for nonzero $a,b$. Then $ba\neq 0$. But also $(ba)^2=0$, contradicting the hypothesis. Therefore such $a,b$ do not exist.
