Clear rings and clear elements

Abstract

An element of a ring $R$ is called clear if it is a sum of a unit-regular element and a unit. An associative ring is clear if each of its elements is clear. In this paper we defined clear rings and extended many results to a wider class. Finally, we proved that a commutative Bezout domain is an elementary divisor ring if and only if every full $2\times 2$ matrix over it is nontrivially clear.

Keywords

• bezout domain
• clean element
• unit-regular element
• full matrix
• elementary divisor ring
• clear element
• clear ring