On rings with the 2-UNJ property

In this paper, we introduce a new class of rings calling them 2-UNJ rings, which generalize the well-known 2-UJ, 2-UU and UNJ rings. Specifically, a ring [Formula: see text] is called 2-UNJ if, for every unit [Formula: see text] of [Formula: see text], the inclusion [Formula: see text] holds, where [Formula: see text] is the set of nilpotent elements and [Formula: see text] is the Jacobson radical of [Formula: see text]. We show that every 2-UJ, 2-UU or UNJ ring is 2-UNJ, but the converse does not necessarily hold, and we also provide counter-examples to demonstrate this explicitly. We, moreover, investigate the connections between these rings and other algebraic properties such as being potent, tripotent, regular and exchange rings, respectively. In particular, we thoroughly study some natural extensions, like matrix rings and Morita contexts, obtaining new characterizations that were not addressed in previous works. Furthermore, we establish conditions under which group rings satisfy the 2-UNJ property. These results not only provide a better understanding of the structure of 2-UNJ rings, but also pave the way for future intensive research in this area. In addition, our achievements considerably improve on recent results given by Tin, On 2-UNJ rings, Asian-European J. Math. 19 (2026) 2550105.