Rings whose non-units are products of torsion units and nilpotents

We prove that every non-invertible matrix of size n ≥ 1 over a division ring can be expressed as the product of an invertible matrix of order at most n and a nilpotent matrix. Motivated by this result, we introduce the notion of a (bounded) TN-ring; that is, a ring in which every non-unit can be written as the product of a torsion unit (of bounded order) and a nilpotent element. We then investigate the behavior of this property under various ring extensions, including group rings and matrix rings. Finally, we provide an example demonstrating that the class of bounded TN-rings is a proper subclass of the class of UN-rings, that is, rings in which every non-unit is expressible as the product of a unit and a nilpotent element.