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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 ...
I had this idea: Denote with $\pi:Mod_R\twoheadrightarrow Ho(Mod_R)$ the projection-localization functor to the homotopy category. Since $0\rightarrowtail M\twoheadrightarrow 0$ every module is both fibrant and cofibrant (or bifibrant), and thus $Ho(Mod_R)$ is isomorphic (not just equivalent) to the classical homotopy category of $Mod_R$ (I guess you read ...