Rings whose non-units are a unit multiple of a quasinilpotent

In this paper, we introduce a class of rings, called [Formula: see text]-rings, defined as rings in which every non-unit element is a product of a unit and a quasinilpotent. These rings properly contain the class of [Formula: see text]-rings, which were investigated more recently in [S. Saini and D. Udar, [Formula: see text]-rings, J. Algebra Appl., accepted]. The [Formula: see text]-rings themselves form a broader class than the so-called [Formula: see text]-rings introduced by Cǎlugǎreanu in [G. Cǎlugǎreanu, UN-rings, J. Algebra Appl. 15(9) (2016) 1650182]. We establish several structural results concerning [Formula: see text]-rings and investigate certain ring extensions, such as group rings and matrix rings. Specifically, we show that if [Formula: see text] is a commutative ring and [Formula: see text] is a locally finite group, then the group ring [Formula: see text] is a [Formula: see text]-ring if and only if it is local. Furthermore, we prove that matrix rings over elementary divisor [Formula: see text]-rings are [Formula: see text]-rings.