Representability of matrices over commutative rings as sums of two potent matrices

We propose some general approach to studying the problem for the representability of every element $a$ in a field $F$ in the form $a = f + g$, with $f^{q_{1}} = f$ and $g^{q_{2}} = g$, where $q_1, q_2 > 1$ are fixed naturals, to imply the analogous representability of every square matrix over $F$. As an application, we describe the fields and commutative rings with $2 \in U(R)$ such that every square matrix over them is the sum of a $q_{1}$-potent matrix and a $q_{2}$-potent matrix for some small values of $q_{1}$ and $q_{2}$.