Let $R$ be a ring and let $S=\{1,s,s^2,s^3,\dots\}$ be a multiplicative system of $R$. Consider the canonical map $R\rightarrow S^{-1}R$. Is $S^{-1}R$ a finitely generated algebra over $R$? It looks like $\frac{1}{s}$ will generate $S^{-1}R$ over $R$.
If $P$ is a prime ideal of $R$, is $R_{P}$ a finitely generated algebra over $R$?
If instead of $S=\{1,s,s^2,s^3,\dots\}$ we take any arbitrary multiplicative system, is $S^{-1}R$ a finitely generated algebra over $R$?
If $S$ is finitely generated as a monoid, then $S^{-1} R$ is finitely generated as an $R$-algebra. In fact, if $S$ is generated by $s_1,\dotsc,s_n$, then $S^{-1} R$ is generated by $\frac{1}{s_1},\dotsc,\frac{1}{s_n}$. If $S$ is not finitely generated, then usually $S^{-1} R$ is not finitely generated. This applies in particular to the case that $S=R \setminus \mathfrak{p}$ for some prime ideal $\mathfrak{p}$. For instance, $\mathbb{Q}$ is not a finitely generated $\mathbb{Z}$-algebra.