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

TL;DR

This paper generalizes classical results on integral domains to their $S$-versions, establishing an $S$-version of Krull's intersection theorem and exploring properties of $S$-fields and $S$-integral domains.

Contribution

It introduces the $S$-version of Krull's intersection theorem and characterizes $S$-fields and $S$-integral domains, extending classical ring theory results.

Findings

Established the $S$-version of Krull's intersection theorem.

Proved that localization of an $S$-field is a $(S)$-field.

Showed that finite $S$-integral domains are $S$-fields.

Abstract

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 $\phi(S)=\left \{\dfrac{s}{1}| \ s\in S\right \}$ 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.