So I'm reading https://arxiv.org/abs/1509.04204, and I'm confused about a certain definition:
Matrix factorizations. Let $S$ be a commutative ring and $x$ and element of $S$. A matrix factorization $(F,g,\phi,\psi)$ of $x$ is a diagram $$F\stackrel{\phi}{\longrightarrow} G\stackrel{\psi}{\longrightarrow}F$$ in which $F$ and $G$ are finitely generated free $S$-modules, and $\phi$ and $\psi$ are $S$-homomorphisms satisfying $$\psi\circ \phi=x\cdot 1_F$$ $$\phi\circ \psi=x\cdot 1_G.$$
How can a $S$-homomorphism from a module to another module be defined? Can you assume that the modules are necessarily $S$?
Here "$S$-homomorphism" means "homomorphism of $S$-modules"; this simply means that $\phi(sa+b)=s\phi(a)+\phi(b)$ for all $s\in S$ and all $a,b\in F$ (and similarly for $\psi$).