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arXiv · 2405.16022

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

Handan Kose·Burcu Ungor·Abdullah Harmanci
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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.

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Handan Kose, Burcu Ungor, Abdullah Harmanci. 2024-05-25. $e$-Reduced rings in terms of the Zhou radical. https://arxiv.org/abs/2405.16022

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