Rings, Matrices over Which Are Representable As the Sum of Two Potent Matrices

Abstract

This paper investigates conditions under which representability of each element \(a\) from the field \(P\) as the sum \(a = f + g\) , where \({{f}^{{{{q}_{1}}}}} = f\) , \({{g}^{{{{q}_{2}}}}} = g\) , and \({{q}_{1}},{{q}_{2}}\) are fixed natural numbers >1, implies a similar representability of each square matrix over the field \(P\) . We propose a general approach to solving this problem. As an application we describe fields and commutative rings where 2 is a unit, over which each square matrix is the sum of two 4-potent matrices.