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

Question

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.

Answer 1

Subrings of domains are domains. Suppose $R$ is a domain and $S$ is a subring. We have to prove every $r\in S\neq 0$ is not a zero divisor, notice that $r$ is not a zero divisor in $R$, so it is not zero divisor in $S$ either, we are done.

In particular notice that division rings are domains.

Answer 2

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

Answer 3

A subring of a division ring may not be a division ring. But a finite subring of a division ring is a division ring because a finite domain is a division ring and subring of a division ring is a domain. Fix $s \in S$, we can consider left translation $sS=\{sh,h \in S\}$, then $sS=S$ because of the cancelation of a domain, so the inverse of $s$ belongs to $S$.