# RIng that is flat over a subring as a right module but not as a left module

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

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

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 $$(x)$$).
• Sure, it just gets annoying since your subring can't be a PID. I think something like the free associative algebra generated by $x, y, z$ subject to $xy=yx$ and $xz=y$ works (with subring $k[x, y]$) (but this is much larger than my original example and I haven't carefully checked it) Oct 13 at 17:52