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.
There's an elementary proof that if $\Omega$ is a self-equivalence of $\underline{C}$ then $C$ also has enough injectives, and projectives and injectives coincide. In particular, this shows that if $C$ is $\text{Mod-}R$ for a ring $R$, then $R$ is quasi-Frobenius.
For any $C$ with enough projectives, it's straightforward to prove that if $X$ and $Y$ are isomorphic in $\underline{C}$, then $X$ is a summand of $Y\oplus P$ and $Y$ is a summand of $X\oplus Q$ in $C$ for some projectives $P$ and $Q$. In particular $X$ is a subobject of a projective iff $Y$ is, and the only objects isomorphic in $\underline{C}$ to zero are the projectives.
So if $\Omega$ is a self-equivalence of $\underline{C}$ then every object of $C$ must be a subobject of a projective.
If $P$ is projective and $\alpha:P\to X$ a monomorphism, then composing with a monomorphism from $X$ to a projective $Q$ and taking the cokernel, we get a short exact sequence $$0\to P\to Q\to Y\to0.$$ Since $\Omega Y\cong P\cong0$ in $\underline{C}$ and we’re assuming $\Omega$ is a self-equivalence, $Y$ is projective, and the short exact sequence splits. Hence $\alpha$ splits, and so $P$ is injective.
Therefore all projectives are injective, and since every object is a subobject of a projective, there are enough injectives.
Every injective is a subobject, and therefore a summand, of a projective, and so every injective is projective.
The classical thing to do is take $R$ to be quasi-Frobenius. Then, the stable category is triangulated, hence $\Omega^1$ is an equivalence. This is in Happel's book Triangulated categories in the representation theory of finite dimensional algebras, though I learned it from Mark Hovey's book Model Categories.
Hovey proved a generalization in Cotorsion pairs, model category structures, and representation theory, to the settings of Gorenstein rings $R$, but you have to kill all modules of finite projective dimension, not just the projective modules. Later, Daniel Bravo generalized it to all rings, in his thesis, and the strongest result appears in The stable module category of a general ring, by Bravo, Gillespie, and Hovey. This involves the class of modules admitting a projective resolution by finitely generated projectives.
