In D & F, theorem $31(2)$ on page $442$, in the proof of the theorem given below (assuming that $M$ is an $R$-module over a commutative ring $R$ with $1$)
If $A$ is any $R$-algebra and $\varphi:M \to A$ is an R-module homomorphism, then there is a unique R-algebra homomorphism $\phi:\mathcal{T}(M) \to A$ such that $\phi|_{M} = \varphi$
they start by saying: "To prove $(2)$, assume that $\varphi:M \to A$ is an $R$-algebra homomorphism."
But why should we assume this? Does not this require $M$ to be an $R$-algebra? $M$ is by assumption just an $R$-module. This is perhaps a naive question, but I don´t understand what motivates this assumption.
