<i>UN</i>-rings

Abstract

A nonzero ring is called a UN-ring if every nonunit is a product of a unit and a nilpotent element. We show that all simple Artinian rings are UN-rings and that the UN-rings whose identity is a sum of two units (e.g. if 2 is a unit), form a proper class of 2-good rings (in the sense of P. Vámos). Thus, any noninvertible matrix over a division ring is the product of an invertible matrix and a nilpotent matrix.