Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures)Khayyam Journal of Mathematics2423-47889120230101Fully $S$-idempotent modules15316116449110.22034/kjm.2022.362523.2664ENFaranak FarshadifarDepartment of Mathematics, Farhangian University, Tehran, Iran0000-0001-7600-994XJournal Article20220917Let $R$ be a commutative ring with identity, $S$ be a multiplicatively closed subset of $R$, and $M$ be an $R$-module. A submodule $N$ of $M$ is said to be \emph{idempotent} if $N=(N:_RM)^2M$. Also, $M$ is said to be \emph{fully idempotent} if every submodule of $M$ is idempotent. The aim of this paper is to introduce the notion of fully $S$-idempotent modules as a generalization of fully idempotent modules and investigate some properties of this class of modules.https://www.kjm-math.org/article_164491_aa3a414cd8221acda43f8775017db403.pdf