Rings in which elements are the sum of a unit and a non-unit

Abstract

We introduce a class of rings, called Harman rings, which is a proper subclass of semiprimitive rings. Let $R$ be a ring with identity. Then $R$ is called Harman if every non-zero element is the sum of a unit and a non-unit in $R$. We investigate relations between Harman rings and some important classes of rings, such as semisimple rings, fine rings, unit-fusible rings, special clean (equivalently, unit-regular) rings and Boolean rings. In particular, we prove that $R$ is a local Harman ring if and only if $R$ is a division ring. A notable result states that for every positive integer $n$, the matrix ring $M_n(R)$ is Harman if and only if $R$ is Harman. This implies, in particular, that any non-zero square matrix over a division ring can be expressed as the sum of a unit matrix and a non-unit matrix.