%0 Journal Article %T On $S$-finite conductor rings %J Khayyam Journal of Mathematics %I Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures) %Z 2423-4788 %A Anebri, Adam %A Mahdou, Najib %A Zahir, Youssef %D 2023 %\ 01/01/2023 %V 9 %N 1 %P 116-126 %! On $S$-finite conductor rings %K $S$-Finite conductor %K Finite conductor %K $S$-Coherence %K $S$-Finite %R 10.22034/kjm.2022.341713.2540 %X Let $R$ be a commutative ring with nonzero identity and $S \subseteq R$ be a multiplicatively closed subset of $R$. In this paper, we introduce and study $S$-finite conductor rings. $R$ is said to be an $S$-finite conductor ring if $(0:a)$ and $Ra\cap Rb$ are $S$-finite ideals of $R$ for each $a,b\in R.$ Some basic properties of $S$-finite conductor rings are studied. For instance, we give necessary and sufficient conditions for a ring to be $S$-finite conductor. Also, we prove that every pre-Schreier $S$-finite conductor domain is an $S$-$GCD$ domain and the converse is true for some particular cases of $S$. Further, we examine the stability of these rings in localization and study the possible transfer to direct product, trivial ring extension and amalgamated algebra along an ideal. %U https://www.kjm-math.org/article_164488_4356630d77982e5912289f4bc1250d61.pdf