Rings in Which Every Quasi-nilpotent Element is Nilpotent

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

1. Chen, H., Sheibani, M., Strongly weakly nil-clean rings, J. Algebra Appl., 16(12)(2017).
2. Cui, J., Chen, J., Characterizations of quasipolar rings, Commun. Algebra, 41(2013), 3207–3217.
3. Danchev, P.V., Lam, T.Y., Rings with unipotent units, Publ. Math. (Debrecen), 88(3-4)(2016), 449–466.
4. Danchev, P.V., Weakly UU rings, Tsukuba J. Math., 40(1)(2016), 101–118.
5. Danchev, P.V., Javan, A., Hasanzadeh, O., Moussavi, A., Rings with u − 1 quasinilpotent for each unit u, J. Algebra Appl.
6. Harte, R.E., On quasinilpotents in rings, Panam. Math. J., 1(1991), 10–16.
7. Karabacak, F., Koşan, M.T., Quynh, T.C., Tai, D.D., A generalization of UJ-rings, J. Algebra Appl., 20(2021).
8. 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.

9. Lam, T.Y., A First Course in Noncommutative Rings, New York, NY, USA: Springer-Verlag, 1991.
10. 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).