On S-Zariski topology

Abstract

Let R be a commutative ring with nonzero identity and, $S\subseteq R$ be a multiplicatively closed subset. An ideal P of R with $P\cap S=\varnothing$ is called an S-prime ideal if there exists an (fixed) $s\in S$ and whenver $ab\in P$ for $a,b\in R$ then either $sa\in P$ or $sb\in P.$ In this article, we construct a topology on the set Spec_S(R) of all S-prime ideals of R which is generalization of prime spectrum of R. Also, we investigate the relations between algebraic properties of R and topological properties of Spec_S(R) like compactness, connectedness and irreducibility.

Keywords:

• Prime spectrum
• S-Zariski topology
• Zariski topology

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13A99
• 54B99

Acknowledgement

The authors would like to thank the referee for his/her great efforts in proofreading the manuscript.