Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes\_R -$ from left $R$-modules to abelian groups that preserves and reflects exact sequences. Faithful flatness for a left $R$-module is defined analogously.


What is an example of an $R$-bimodule that is faithfully flat as a right module, but not faithfully flat as a left module? Or what is an example of an $R$-bimodule that is faithfully flat as a left module, but not faithfully flat as a right module?



Faithful flatness for rings
Let $G$ be a non-trivial finite group and let $R=\mathbb ZG$. We can view $M=\mathbb ZG$ as an $R$-bimodule via the right regular module structure on the right and the trivial module structure on the left (so left multiplication by $g\in G$ fixes $M$). Then $M$ is faithfully flat as a right module because $M\otimes\_R ()$ is the underlying abelian group functor. But $M$ is not flat as a left $R$-module. Indeed $M$ is finitely presented (cf. the bar resolution) and so if it were flat it would be projective. But then the trivial module $\mathbb Z$ would be projective which is never the case for a non-trivial finite group as no idempotent $e$ satisfies $ge=e$ for all $g\in G$ in a group algebra unless the order of $G$ is a unit in the coefficient ring.


Of course you can switch the roles of right and left.


