On integral domains in which every ascending chain on principal ideals is $S$-stationary

Bull. Korean Math. Soc. Published online July 9, 2020

Ahmed Hamed and Hwankoo Kim
University of Monastir, Hoseo University

Abstract : Let $D$ be an integral domain and $S$ a multiplicative subset of $D.$ An ascending chain $(I_k)_{k\in\mathbb{N}}$ of ideals of $D$ is said to be $S$-stationary if there exist a positive integer $n$ and an $s\in S$ such that for each $k\geq n,$ $sI_k\subseteq I_n.$ As a generalization of domains satisfying ACCP (resp., ACC on $*$-ideals) we define $D$ to satisfy $S$-ACCP (resp., $S$-ACC on $*$-ideals) if every ascending chain of principal ideals (resp., $*$-ideals) of $D$ is $S$-stationary. One of main results of this paper is the Hilbert basis theorem for an integral domain satisfying $S$-ACCP. Also we investigate the class of such domains $D$ and we generalize some known related results in the literature. Finally some illustrative examples regarding the introduced concepts are given.