>Let $C$ be an abelian category with enough projectives and $\underline{C}$ the stable category of $C$ that is obtained by factoring out projective modules.
When is the functor $\Omega^1 : \underline{C} \rightarrow \underline{C}$ an equivalence?

We can assume $C$ is a module category of a ring $R$ in case that helps. For $R$ a finite dimensional algebra, this should be true iff $R$ is self-injective.

>Is there an explicit example of a ring $R$ which is not quasi-Frobenius and has the property that $\Omega^1$ is an equivalence on the stable module category?