I have the following definitions for commutative rings $R\subseteq S$.
$R[x]$ is the polynomial ring of polynomials with coefficients from $R$.
For $a\in S$, $R[a] = \{f(a)): f\in R[x]\}$ and $R(a)$ is the smallest field containing $a$ and $R$ (I guess here one would need $S$ to be a field).
Now I have also learnt about quotient fields. In particular $R(x) = \{f(x)/g(x): f,g\in R[x], g\neq 0\}$ is the quotient field of $R[x]$.
Given the notation my question is: Is $R(a)$ then the quotient field of $R[a]$?
As you have noted parenthetically, you want to assume that $S$ is a field (or at least an integral domain). In that case, you actually get $$Q(R[a])=Q(R)(a).$$