Duo property for rings by the quasinilpotent perspective

TL;DR

This paper introduces the concept of qnil-duo rings, a new generalization of commutativity based on quasinilpotent elements, and explores their properties and implications in ring theory.

Contribution

It defines and studies right and left qnil-duo rings, showing their properties and asymmetry, and connects these concepts to other ring classes like exchange rings.

Findings

Right qnil-duo rings are abelian.

If the Hurwitz series ring is right qnil-duo, then the base ring is right qnil-duo.

A right qnil-duo exchange ring has stable range 1.

Abstract

In this paper, we focus on the duo ring property via quasinilpotent elements which gives a new kind of generalizations of commutativity. We call this kind of ring qnil-duo. Firstly, some properties of quasinilpotents in a ring are provided. Then the set of quasinilpotents is applied to the duo property of rings, in this perspective, we introduce and study right (resp., left) qnil-duo rings. We show that this concept is not left-right symmetric. Among others it is proved that if the Hurwitz series ring $H(R; \alpha)$ is right qnil-duo, then $R$ is right qnil-duo. Every right qnil-duo ring is abelian. A right qnil-duo exchange ring has stable range 1.