On S-Integral Domains and S-Version of Krull Intersection Theorem

Let $S\subseteq R$ be a multiplicatively closed subset of a ring $R$. We extend several results on integral domains to their $S$-versions and establish the $S$-version of Krull intersection theorem. We also show that if $R$ is an $S$-field, then the localization of $R$ with respect to $S$ is a $\phi (S)$-field, where $\phi (S)=\{{\displaystyle \frac{s}{1}}\mathrm{\mid }s\in S\}$ is a multiplicatively closed subset of $S^{-1}R$, and prove the converse under the condition of finiteness of $S$. As a consequence, we show that every finite $S$-integral domain is an $S$-field. Also, we provide several examples to illustrate the significance of our findings.