Bull. Korean Math. Soc. 2024; 61(1): 83-92
Online first article January 22, 2024 Printed January 31, 2024
https://doi.org/10.4134/BKMS.b230023
Copyright © The Korean Mathematical Society.
Bayram Ali Ersoy, Ünsal Tekir, Eda Yıldız
Yildiz Technical University; Marmara University; Yildiz Technical University
Let $R$ be a commutative ring with nonzero identity and $M$ be an $R$-module. In this paper, we first introduce the concept of $S$-idempotent element of $R$. Then we give a relation between $S$-idempotents of $R$ and clopen sets of $S$-Zariski topology. After that we define $S$-pure ideal which is a generalization of the notion of pure ideal. In fact, every pure ideal is $S$-pure but the converse may not be true. Afterwards, we show that there is a relation between $S$-pure ideals of $R$ and closed sets of $S$-Zariski topology that are stable under generalization.
Keywords: Prime spectrum, Zariski topology, $S$-Zariski topology, pure ideal, $S$-idempotent, Stone type theorem
MSC numbers: Primary 13A15, 13A99, 54B99
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