On S-integral domains and S-version of Krull intersection theorem

Let S ⊆ 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 ϕ(S)-field, where ϕ(S) ={s/1| s ∈ 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.