Let C be a finite field and let Cₘ denote the ring consisting of all m × 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 Cₘ is the range of a polynomial without constant term if and only if 0 ∈ A and uAu- 1 ⊆ A for all invertible elements u ∈ Cₘ.
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Chen-Lian Chuang (1990) studied this question.