Commuting maps on idempotents and functional identities on finite subsets of fields

Abstract

Let F be a field, n ≥ 3 and let M n be the ring of all n × n matrices with entries in F . For a given subset H of M n , consider H ¯ = H ∪ V with V = { α E ii ∣ α ∈ F and E ii is a matrix unit for each i = 1 , … , n } . In this paper, under a mild technical assumption on F , we describe additive maps G : M n → M n satisfying [ G ( X ) , X ] = G ( X ) X − XG ( X ) = 0 for all X ∈ H ¯ in the following settings: H = E = { A ∈ M n ∣ A is idempotent } ; H = S = { A ∈ M n ∣ A is algebraic of degree 2 } ; H = P = { X ∈ M n ( F ) ∣ tr ( X ) ∈ L } where L is the prime field of F . These maps are so-called commuting on H ¯ . Our findings will allow us to conclude that for H = S the map G has the so-called standard form. Moreover, we will show that G is commuting on E ¯ if and only if G is commuting on P ¯ . Our investigation will lead us to consider functional identities on F of the form H 1 ( r ) s + H 2 ( s ) r = 0 with r, s belonging to a 2-point subset of F . At the end, we will discuss about the existence of non-standard additive commuting maps on E ¯ which are not commuting on the set of rank 1 matrices and vice versa.