Let $R \subseteq S$ be an extension of rings with unit. Suppose that $S$ is free as left $R$-module. I wonder what can said about the freeness of $S$ as right $R$-module. To be a little more precise let's consider the following questions:

Does someone know an example such that $S$ is free as left $R$-module, but isn't free as right $R$-module.

What are conditions that imply that $S$ is free as left module iff it's free as right module ?

What are categorial examples such that $S$ is free as left as well as right $R$-module ?

An examples for 2. is

- $R$ is a subring of the center of $S$

and examples for 3. are:

$R \subseteq S$ is a Frobenius extension

$S$ is a Hopf algebra over a field and $R$ a sub-Hopf algebra (freeness holds by the Nichols-Zoeller theorem)