On ranges of polynomials in finite matrix rings

Abstract:

Let $C$ be a finite field and let ${C_m}$ denote the ring consisting of all $m \times m$ matrices over $C$. By a polynomial, we mean a polynomial in noncommuting indeterminates with coefficients in $C$. It is shown here that a subset $A$ of ${C_m}$ is the range of a polynomial without constant term if and only if $0 \in A$ and $uA{u^{ - 1}} \subseteq A$ for all invertible elements $u \in {C_m}$.