What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module.


The following question is my motivation:


[Faithful flatness for rings](https://mathoverflow.net/questions/403926/faithful-flatness-for-rings)


But please note that I am looking for a more specific counterexample.



RIng that is flat over a subring as a right module but not as a left module
Take the associative algebra over a field $k$, with generators $x$ and $y$ subject to the relation $xy=0$. This admits a basis consisting of monomials of the form $y^a x^b$. It thus contains a subring $k[x]$, and is flat (even free) over this subring as a right module, on the basis $y^a$, $a\geq 0$. As a left module, it isn't flat, since it isn't even torsion free (for example, $y$ is annihilated by $(x)$).


