Commuting maps on rank-k matrices

Abstract

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 à n matrices over an arbitrary field K with center Z. Fix a natural number 2â¤sâ¤n. If G:Mn(K)âMn(K) is an additive map such that G(x)x=xG(x) for every rank-s matrix x â Mn(K), then there exist an element λ â Z and an additive map μ : Mn(K) â Z such that G(x) = λx + μ(x) for each x â Mn(K).