$e$-Reduced rings in terms of the Zhou radical

TL;DR

This paper explores the concept of Zhou radical in rings to define and analyze the properties of Zhou right and left e-reduced rings, providing examples, properties, and extensions related to these structures.

Contribution

It introduces the notion of Zhou right (left) e-reduced rings, investigates their properties, and provides examples and extensions, linking them to other ring classes.

Findings

Zhou right e-reduced rings include right e-semicommutative, e-symmetric, central semicommutative, and weak symmetric rings.

Full matrix rings are not necessarily Zhou right e-reduced, but certain subrings are.

The paper establishes properties and examples of Zhou right e-reduced rings and their extensions.

Abstract

Let $R$ be a ring, $e$ an idempotent of $R$ and $\delta(R)$ denote the intersection of all essential maximal right ideals of $R$ which is called Zhou radical. In this paper, the Zhou radical of a ring is applied to the $e$-reduced property of rings. We call the ring $R$ {\it Zhou right} (resp. {\it left}) {\it $e$-reduced} if for any nilpotent $a$ in $R$, we have $ae\in \delta(R)$ (resp. $ea\in \delta(R))$. Obviously, every ring is Zhou $0$-reduced and a ring $R$ is Zhou right (resp., left) $1$-reduced if and only if $N(R)\subseteq \delta(R)$. So we assume that the idempotent $e$ is nonzero. We investigate basic properties of Zhou right $e$-reduced rings. Furthermore, we supply some sources of examples for Zhou right $e$-reduced rings. In this direction, we show that right $e$-semicommutative rings (and so right $e$-reduced rings and $e$-symmetric rings), central semicommutative rings and weak symmetric rings are Zhou right $e$-reduced. As an application, we deal with some extensions of Zhou right $e$-reduced rings. Full matrix rings need not be Zhou right $e$-reduced, but we present some Zhou right $e$-reduced subrings of full matrix rings over Zhou right $e$-reduced rings.