Degeneracy of some derived categories

We study derived categories for the category of the modules over some generalized rings. In particular, the cases of $\mathcal O_\mathbb R$ and of $\mathbb F_{1^n}$ are considered. It is shown that these derived categories are degenerate. The degeneracy means that every isomorphism in such a category can be detected on the $\pi_0$- and $\pi^0$-levels.