Let $R$ be an associative ring. Let $K(R)$ be the category of chain complexes of $R$-modules and chain homotopy classes of maps between them, and let $D(R)$ be its localization with respect to acyclic complexes.


Is there a ring $S$ such that $K(R) = D(S)$? Is the question more likely to have a positive answer if I allow $S$ to be a differentially graded ring, or don't require that $S$ has an identity?



Is the homotopy category of a ring also the derived category of another ring?
Eric is right, $K(\mathbb{Z})$ has no generating set. This is Lemma E.3.2 in Neeman's *Triangulated Categories*. Presumably, the proof will apply to many other rings, but I can't comment on that.


