Let $A$ be an $R$-algebra over a commutative ring $R$.
Under appropriate finiteness conditions (A finitely generated as $R$-module, $R$ Noetherian), any finitely generated $A$-bimodule $M$ is known to have a free resolution, that is, a complex $$ \ldots (A \otimes_{R} A)^{r_2} \to (A \otimes_{R} A)^{r_1} \to (A \otimes_{R} A)^{r_0} $$ of $A$-bimodules with non-zero integers $r_i$ and only one non-vanishing homology, which is at zero degree and isomorphic to $M$.
Are there any "good" necessary or sufficient criteria known for the bimodule $M$ to have a "naive" resolution of the form $$ \ldots A^{r_2} \to A^{r_1} \to A^{r_0}$$ viewing $A$ as $A$-bimodule in a natural way?
Does the category of such bimodules have any notable properties?
To avoid boredom, let us assume that $A$ is not just the base ring $R$.
Any answer or mention of reference will be appreciated. Thank you for reading!