Rings in Which Every Quasi-nilpotent Element is Nilpotent

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.