I'm studying homotopical algebra and I'm trying to prove the following fact:
If $R$ is a quasi-Frobenius ring (for $R$-modules one has "projective module$\iff$ injective module") then the stable category of $R$-modules is (equivalent to) the homotopy category of $R$-mod, once the latter has been equipped with the model structure in which cofibrations are monomorphisms, fibrations are epimorphisms and weak equivalences are stable-isomorphisms. In order to do this, since in this model structure each module is both fibrant and cofibrant, it's enough to show that the homotopy relation is the same as the stable-equivalence relation. More precisely one needs to show that $f$ is left (or right, it's the same) homotopic to $g$ $\iff$ $f$ is stable equivalent to $g$.
Now my problem is in the "$\Leftarrow$" part: how can I construct a left, say, homotopy between $f$ and $g$ once I know that $f-g$ factorizes through a projective (hence also injective) $R$-module? Thanks in advance.