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

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

References

• Chen, H., Sheibani, M., Strongly weakly nil-clean rings, J. Algebra Appl., 16(12)(2017).
• Cui, J., Chen, J., Characterizations of quasipolar rings, Commun. Algebra, 41(2013), 3207–3217.
• Danchev, P.V., Lam, T.Y., Rings with unipotent units, Publ. Math. (Debrecen), 88(3-4)(2016), 449–466.
• Danchev, P.V., Weakly UU rings, Tsukuba J. Math., 40(1)(2016), 101–118.
• Danchev, P.V., Javan, A., Hasanzadeh, O., Moussavi, A., Rings with u − 1 quasinilpotent for each unit u, J. Algebra Appl.
• Harte, R.E., On quasinilpotents in rings, Panam. Math. J., 1(1991), 10–16.
• Karabacak, F., Koşan, M.T., Quynh, T.C., Tai, D.D., A generalization of UJ-rings, J. Algebra Appl., 20(2021).
• Kos¸an, T., Wang, Z., Zhou, Y., Nil-clean and strongly nil-clean rings, Journal of Pure and Applied Algebra, 220(2)(2016), 633–646.
• Lam, T.Y., A First Course in Noncommutative Rings, New York, NY, USA: Springer-Verlag, 1991.
• Tien, N.Q., Koşan, M.T., Quynh, T.C., Remarks on rings with u − 1 quasi-nilpotent for each unit u, J. Algebra Appl., (2025).