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$S$-versions and $S$-generalizations of idempotents, pure ideals and Stone type theorems
Bayram Ali Ersoy,Unsal Tekir,Eda Yildiz 대한수학회 2024 대한수학회보 Vol.61 No.1
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.