# On $S$-finite conductor rings

Document Type : Original Article

Authors

1 Laboratory of Modelling and Mathematical Structures; Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco

2 Laboratory of Modelling and Mathematical Structures \\Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco

3 Laboratory: Mathematics, Computing and Applications- Information Security (LabMiA-SI); Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco

10.22034/kjm.2022.341713.2540

Abstract

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.

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