Rings in Which Every Quasi-nilpotent Element is Nilpotent

Year 2025, Volume: 17 Issue: 1, 75 - 81, 30.06.2025

Tin Phan Hong

https://doi.org/10.47000/tjmcs.1675857

https://izlik.org/JA37RE48JS

Abstract

A ring \( R \) is called a QN-ring if \( R \) satisfies the equation \( Q(R) = N(R) \). In this paper, we present some fundamental properties of the class of QN-rings. It is shown that for \( R \) being a 2-primal (nil-semicommutative) ring, \( R \) is a QN-ring if and only if \( Q(R) \) is a nil ideal; if \( R \) is a QN-ring, then \( R/J(R) \) is a semiprime ring; if \( R \) is a QN-ring and \( R/J(R) \) is nil-semicommutative, then \( R \) is a feckly reduced ring. We also show that if $T_n(R, \alpha)$ is a QN-ring, then $R$ is a QN-ring.

Keywords

QN-ring , UU ring , NI-ring , NJ-ring

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