On ranges of polynomials in finite matrix rings

Abstract

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