Multiplication by homotopy classes is module-linear

Observation 8.4.2. Multiplication by \(s \in \pi _d(R)\) is the natural transformation of functors \(\Mod _R \to \Mod _R\) given by \[ s\colon M[d] \simeq R[d]\otimes _R M \xrightarrow {\ s \otimes \id \ } R \otimes _R M \simeq M. \] On homotopy groups this is the map \(x \mapsto s\cdot x\).