What tools are available for constructing various (noncommutative) domains with a given subdomain? That is, how can I begin to examine various domains which contain the domain $S$, other than obviously $S$ itself?
If we needed only a ring $R$ with a subring $S$, we could take direct sums of $S$ with other rings, but unfortunately the direct sum does not preserve the domain property, so this construction method is not useful.
In the more familiar commutative case, you could look at certain quotients of polynomial rings over $S$. The quotient $S[x]/I$ will contain a copy of $S$ so long as $I$ trivially intersects the zeroth graded component of $S[x].$ Furthermore, if $S$ is commutative and unital, $S[x]/I$ will be a domain if and only if $I$ is a prime ideal. This will give you many examples of commutative domains that have $S$ as a subring.
This recipe only has to be fudged a little bit to allow for noncommutative quotient rings. Instead of starting with a polynomial ring over $S$, start with a free algebra over $S$ like $S\langle x,y\rangle$ instead. Then the quotient $S\langle x, y \rangle/I$ will be a domain if and only if $I$ is a completely prime ideal.