Erdman's linear algebra text: http://web.pdx.edu/~erdman/ has this:
Background: $\operatorname{Hom}(V)$ is a unital ring under operations $f+g\colon V\to V\colon x\mapsto f(x)+g(x)$ and composition $gf\colon V\to V\colon x\mapsto g(f(x))$. A function $f\colon R\to S$ between unital rings is a unital ring homomorphism if $f(x+y)=f(x)+f(y)$, $f(xy)=f(x)f(y)$, and $f(\mathbf{1}_R)=\mathbf{1}_S$.
I can recover (1)-(5) as $(V,+)$ is an Abelian group but I'm not sure how I recover (6)-(10) from $M$. Specifically, for (6), if $\alpha\in F$ and $\mathbf{x}\in V$, how do I show $\alpha\mathbf{x}\in V$?
What is $\alpha\mathbf{x}$? $M$ takes elements of $F$ only and $\mathbf{x}\notin F$. And since $\alpha$ and $\mathbf{x}$ are elements of rings, they're not functions in $\operatorname{Hom}(V)$, so I can't just compose them.
I am looking for a hint on how to use $M$ to make sense of $\alpha\mathbf{x}$.
What you do is to define $\alpha x$ as $M(\alpha)(x)$. You have that $M(\alpha)$ is a homomorphism of $V$, so you can evaluate it on $x$.
Conversely, if you have scalar multiplication you define $M(\alpha)$ to be the homomorphism $x\longmapsto \alpha x$.