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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.

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    Maybe you just have different definitions, but aren't domains required to be commutative, while division rings need not be? If so, the problem seems false. – 2017-01-31
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    You seem to have confused the title box for the tags box. One is used for categorizing questions, the other is used to inform people briefly what the question is. It's not a good idea to repeat the same information in both places. I have adjusted your title for you. – 2018-06-16

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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.

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