Show that a subring of a division ring must be a domain. 
Show that a subring of a division ring must be a domain.

Let $S$ be a subring of $R$ and let $R$ be a division ring.
Can we just say that since every element has an inverse in $R$, then every element also has an inverse in $S$, then we can deduce that since for all elements in $S$ we can find a non zero inverse element?
Assume by contradiction that there is a zero divisor $k$ in $S$, such that $ks=0$ or $sk=0$ for all $s$ in $S$ well we can find $k$ inverse such that it is not true. Thus contradicting assumption, blah blah blah no zero divisor means $S$ is a domain and fin, drop the mic and such or am I completely off?
I realize I am wrong now because a subring of a division ring does not imply that there is an inverse, right? So I don't know where to go from here.

Hint $\ $ If $\,ax=0\,$ has  unique root $\,x=0\,$ in $R\,$ then the same holds true in every subring.
