NILPOTENT AND LINEAR COMBINATION OF IDEMPOTENT MATRICES

Abstract

A ring $R$ is Zhou nil-clean if every element in $R$ is the sum of a nilpotent and two tripotents. Let $R$ be a Zhou nil-clean ring. If $R$ is of bounded index or 2-primal, we prove that every square matrix over $R$ is the sum of a nilpotent and a linear combination of two idempotents. This provides a large class of rings over which every square matrix has such decompositions by nilpotent and linear combination of idempotent matrices. $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$