This question is about extending modules of fractions to bimodules of fractions. I would not be surprised if the result is known, I have tried looking in Goodearl and Warfield, but may have missed the point as I am not an expert in algebra. The motivation lies in looking at invertible bimodules in non commutative algebraic geometry.
Given sufficient conditions, for a ring $R$ and a multiplicative set of regular elements $X$ we can form the ring of fractions $RX^{-1}$. Now for an right $R$ module $E$, we can form the right $RX^{-1}$ module $E \otimes_R RX^{-1}$.
However suppose that $E$ is actually a $R$-bimodule. When is $E \otimes_R RX^{-1}$ then a $RX^{-1}$ bimodule?
As I see it, this question can be phrased in the same sort of terms as the Ore condition on the ring, that is given $x\in X$ and $e\in E$ is there $y\in X$ and $f\in E$ so that $ey=xf$. Is this the right approach? If so, are there general conditions saying when this construction works? Where can I find out about it?
Apologies for my lack of knowledge here, I hope that this does not sound too trivial to the experts. The idea is to apply it to localisation of bimodules over quantum groups and their homogenous spaces.
Edit: I should say that the reason why I want $E \otimes_R RX^{-1}$ rather than something potentially bigger to be the bimodule is that otherwise taking fractions will not commute with tensor product, or rather we will likely not get a monoiodal functor between bimodule categories.