Let $R$ be a unital ring and $A$ and $B$ unital subrings of $R$ (not necessarily commutative). Then $A$ and $B$ are both bimodules over $A \cap B$ and I was wondering what conditions (if any) have to be met in order for $$A \otimes_{A \cap B} B \to R, a \otimes b \mapsto ab $$ to be an isomorphism onto its image (i.e. injective).
This question came to my mind while thinking about the analogue situation in group theory where $$ A \times_{A \cap B} B \to G, [a,b] \mapsto ab$$ for subgroups $A,B \subseteq G$ is always an isomorphism onto its image $AB$. The group rings and their subrings generated by subgroups are thus examples where the above morphism is an isomorphism onto its image.