Structural results on harmonic rings and lessened rings

TL;DR

This paper applies algebraic and topological methods to study harmonic rings, introducing the concept of lessened rings and providing new structural results and criteria for various classes of rings.

Contribution

It introduces the new notion of lessened rings, generalizes reduced rings, and characterizes their structure and properties within harmonic and Gelfand rings.

Findings

Product of rings is lessened iff each factor is lessened.

Characterization of locally lessened mp-rings.

Finite products of fields, domains, local rings, and lessened rings are characterized.

Abstract

In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring $A$ modulo its Jacobson radical is a zero dimensional ring, then $A$ is a clean ring. It is also proved that, for a given Gelfand ring $A$, then the retraction map Spec$(A)\rightarrow{\rm Max}(A)$ is flat continuous if and only if $A$ modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring $A$, then the retraction map Spec$(A)\rightarrow{\rm Min}(A)$ is Zariski continuous if and only if ${\rm Min}(A)$ is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes "reduced ring" notion. Specially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring when is a finite product of fields, integral domains, local rings, and lessened quasi-prime rings.