k-Potence preserving maps without the linearity and surjectivity assumptions

Let Mn be the space of all n × n complex matrices, and let Γn be the subset of Mn consisting of all n × n k-potent matrices. We denote by Ψn the set of all maps on Mn satisfying A − λB ∈ Γn if and only if ϕ(A) − λϕ(B) ∈ Γn for every A,B ∈ Mn and λ ∈ C. It was shown that ϕ ∈ Ψn if and only if there exist an invertible matrix P ∈ Mn and c ∈ C with ck−1 = 1 such that either ϕ(A) = cPAP−1 for every A ∈ Mn, or ϕ(A) = cPATP−1 for every A ∈ Mn.