It would be nice to have several examples of an interesting $R$-algebra $A$, where $R$ is a non-commutative ring (plausible definitions can be found here).
One example is a polynomial ring over any non-commutative ring, see this question.
Also, I wonder if some properties of commutative ring extensions are still valid for two non-commutative rings, for example, the definition of a separable ring extension $R \subseteq A$, where $R$ is a non-commutative ring (hence $A$ is also non-commutative). The usual definition where $R$ is a commutative ring, can be found in this post (in his notations, is it possible to just replace $k$ by a non-commutative ring? Is there a problem with tensoring over a non-commutative ring?).
Edit: After receiving two comments requesting me to be more precise/pick a definition, my new question is:
Let $R \subseteq A$, $R$ is a non-commutative ring ($R$ and $A$ are associative with $1$). It would be nice to have examples of such rings with $A$ being a flat $R$-module. My example: $R$ is a division ring.
Thank you very much!
This is a quick and dirty way to get a lot of examples:
A ring is von Neumann regular iff all of its modules are flat. So, you can pick any ring $A$ containing a von Neumann regular ring $R$, and you have an example.
A few other obvious constructions that would work for any ring $R$ include regarding $\prod_{i=1}^n R$ and $M_n(R)$ and $R[x]$ as $R$ algebras, since all are free hence flat modules over $R$.