Characterizations of Gelfand rings specially clean rings and their dual rings

TL;DR

This paper introduces new criteria and classes for Gelfand, clean, and mp-rings, including purified rings, and explores their properties and implications in scheme topology.

Contribution

It presents novel characterizations of Gelfand and mp-rings, introduces purified rings, and links topological properties of schemes with ring-theoretic conditions.

Findings

Reduced purified rings are characterized.

Pure ideals of reduced Gelfand and mp-rings are characterized.

Affine opens in Hausdorff schemes are stable under finite unions.

Abstract

In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.