Classical S-Zariski Topology of a Module

Abstract

Let R be a commutative ring with identity and let S be a multiplicatively closed subset of R. A submodule P of an R-module M with $(P:_{R}M)\cap S=\emptyset $ is said to be an S-prime submodule of M if there exists a fixed $s\in S$ and whenever $am\in P$, then $sa\in (P:_{R}M)$ or $sm\in P$ for each $a\in R$, $m\in M$. The set of all S-prime submodules of M is denoted by $Spec_{S}(M)$. In this paper, we construct and investigate a topology on $Spec_{S}(M)$ which we will call classical S-Zariski topology for an R-module M. We use specific algebraic properties of M to obtain some topological properties such as separation axioms, compactness, connectedness, and irreducibility. We also investigate classical S-Zariski topology from the point of view spectral spaces by using Hochster’s characterization.

Keywords

S-prime submodule / S-multiplication module / S-Zariski topology / Classical S-Zariski topology / 13C13 / 13C99 / 54B99

References

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Funding

Trakya University

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