the rank of $M$ is then defined by the dimension of the vector space $S^{-1}M$.

Question

Let $M$ be a $A$-module, where $A$ be the commutative ring with unity. Set $S=A \setminus \{0 \}$ and $K=S^{-1}A$, which is a field. Then $S^{-1}M$ is a $K=S^{-1}A$-vector space. The rank of $M$ is then defined by the dimension of the vector space $S^{-1}M$.

My question-

$(i)$ How to show $S^{-1}A$ is a field ?

$(ii)$ How to show $S^{-1}M$ is a $S^{-1}A$-vector space ?