Let $R$ be a ring with stable module category $C$ and $\Omega^1$ the first syzygy functor.

Question:

Given $d \geq 1$, for which rings is $\Omega^d$ an equivalence of $C$?

For finite dimensional algebras this should be the case iff $R$ is selfinjective, but I do not know much about general rings.

This question seems natural, but I have not found a reference. Perhaps it is trivial (sorry if it is, maybe I shouldnt ask such questions late at night).

You may also think of the situation of an abelian category with enough projectives instead of a ring (and its module category).