Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive? Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos and whose fibrations are epis. It follows that trivial cofibrations are monos with bijective (projective-injective) cokernel and trivial fibrations are epis with bijective kernel. 
Does the homotopy category $Ho(Mod_R)$ inherit at least the preadditive structure from $Mod_R$? I know that the answer must be positive, as the homotopy category is supposed to be isomorphic to the stable category of $Mod_R$ (although I cannot find a proof of that), which is additive.
I tried to define $[f]+[g]:=[f+g]$ and $-[f]:=[-f]$ (where $[f]$ denotes the homotopy class of $f\in Mod_R(X,Y)$), but I'm having troubles proving that this operation is well defined.
I also proved that the class $Null(X,Y)$ of nullhomotopic maps ($f\approx 0$) is a submodule of $Mod_R(X,Y)$, but i still have to show that $f\approx g\Leftrightarrow f-g\in Null(X,Y)$ (which is pretty much the same thing as before) to conclude the proof. 
 A: I'll use the terminology of Hovey's book on Model Categories.
If $X$ is an $R$-module, and $\iota:X\to I$ a monomorphism from $X$ to an injective module, then $X\oplus I$ is a cylinder object for $X$ with the maps
$$X\oplus X\stackrel{\begin{pmatrix}1&1\\\iota&0\end{pmatrix}}{\to}X\oplus I\stackrel{\begin{pmatrix}1&0\end{pmatrix}}{\to}X.$$
All objects are both fibrant and cofibrant, and by Corollary 1.2.6 of Hovey, it doesn't matter which cylinder object is used to define homotopy, so we may as well stick with the one above.
So if $\alpha,\beta:X\to Y$ are two maps, then $\alpha\sim\beta$ if and only if there is a map
$$X\oplus I\stackrel{\begin{pmatrix}h_X&h_I\end{pmatrix}}{\to}Y$$
such that
$$\begin{pmatrix}h_X&h_I\end{pmatrix}\begin{pmatrix}1&1\\\iota&0\end{pmatrix}=\begin{pmatrix}\alpha&\beta\end{pmatrix},$$
i.e., $h_X=\beta$ and $\alpha-\beta=h_I\iota$.
So $\alpha\sim\beta$ if and only if $\alpha-\beta$ factors through $\iota$. In particular, the null-homotopic maps are those that factor through $\iota$ and two maps are homotopic if and only if their difference is null-homotopic.
