Idempotence preserving maps on spaces of triangular matrices

SupposeF is an arbitrary field. Let |F| be the number of the elements ofF. LetT _n (F) be the space of allnxn upper-triangular matrices overF. A map Ψ: T _N (F) → T _N (F) is said to preserve idempotence ifA - λ B is idempotent if and only if Ψ(A) - λΨ(B) is idempotent for anyA, B ∈ T _n (F) and λ ∈ F. It is shown that: when the characteristic ofF is not 2, |F|>3 and n ≥ 3, Ψ:T _n (F) → T _n (F) is a map preserving idempotence if and only if there exists an invertible matrixP τ T _n (F) such that either ?(A) = PAP ^?1 for everyA ∈ T _n (F) or Ψ(A) = PJA ^t JP ^?1 for everyA ∈ T _n (F), whereJ = ∑ _n=1 ⁿ E _i,n+1?i and E_ij is the matrix with 1 in the (i,j)th entry and 0 elsewhere.