Rings, matrices over which are representable as the sum of two potent matrices

This paper investigates conditions under which representability of each element $a$ from the field $P$ as the sum $a= f+g$, with $f^{q_{1}} = f$, $g^{q_{2}} = g$ and $q_1, q_2$ are fixed integers $>1$, 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 with $2$ is a unit, over which each square matrix is the sum of two $4$-potent matrices.