Let n , α ∈ N ≥ 2 and let K be an algebraically closed field with characteristic 0 or greater than n . We show that if f ∈ K [ X ] and A , B ∈ M n ( K ) satisfy [ A , B ] = f ( A ) , then A , B are simultaneously triangularizable. Let R be a reduced ring such that n ! is not a zero divisor and let A be a generic matrix over R ; we show that X = 0 is the sole solution of A X - X A = X α . Let R be a commutative ring with unity; let A be similar to d i a g ( λ 1 I n 1 , … , λ r I n r ) such that, for every i ≠ j , λ i - λ j is not a zero divisor. If X is a nilpotent solution of X A - A X = X α g ( X ) where g ∈ R [ X ] , then A X = X A .