The $S$-flat topology

TL;DR

This paper introduces the $S$-flat topology on the $S$-prime spectrum of a commutative ring, providing algebraic descriptions of its topological properties such as compactness and irreducibility.

Contribution

It defines a new topology on the $S$-prime spectrum and characterizes its topological properties algebraically.

Findings

The $S$-flat topology is well-defined on the $S$-prime spectrum.

Topological properties like compactness and irreducibility are described algebraically.

The topology offers new insights into the structure of $S$-prime spectra.

Abstract

For a commutative ring $R$ with unit $1\ne 0$ and a multiplicatively closed subset $S$ of $R$, we introduce a new topology on the $S$-prime spectrum $\mathrm{Spec}_SR$ of $R$ called the $S$-flat topology. Our aims is to give an algebraic descriptions of the topological properties like compactness, irreducibility, connectivity and noetherianess with respect to this new topology .