Commuting maps, rank-k matrices
In this short note we give a new proof and a slight improvement of the Franca Theorem. More precisely we prove: Let n \geq 3 be a natural number, and let Mn(K) be the ring of all n × n matrices over an arbitrary field K with center Z. Fix a natural number 2≤s≤n. If G:Mn(K)→Mn(K) is an additive map such that G(x)x=xG(x) for every rank-s matrix x ∈ Mn(K), then there exist an element λ ∈ Z and an additive map μ : Mn(K) → Z such that G(x) = λx + μ(x) for each x ∈ Mn(K).
Xu, Xiao-Wei and Yi, Xiaofei.
"Commuting maps on rank-k matrices",
Electronic Journal of Linear Algebra,