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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∈Mn(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 AX-XA=Xα. Let R be a commutative ring with unity; let A be similar to diag(λ1In1,…,λrInr) such that, for every i≠j,λi-λj is not a zero divisor. If X is a nilpotent solution of XA-AX=Xαg(X) where g∈R[X], then AX=XA

The Matrix Equation XA- AX=Xαg(X) over Fields or Rings