Communications of the
Korean Mathematical Society CKMS

The reversible property of rings was initially introduced by Habeb and plays a role in noncommutative ring theory. In this note we study the reversible ring property on nilpotent elements, introducing the concept of {\it commutativity of nilpotent elements at zero} (simply, a {\it CNZ} ring) as a generalization of reversible rings. We first find the CNZ property of $2$ by $2$ full matrix rings over fields, which provides a basis for studying the structure of CNZ rings. We next observe various kinds of CNZ rings including ordinary ring extensions.

Keywords: CNZ ring, reversible ring, matrix ring, polynomial ring, skew Laurent polynomial ring

MSC numbers: 16U80, 16N40