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

Abstract

Let be a field, and let be the ring of all matrices with entries in For a given subset H of consider with In this paper, under a mild technical assumption on we describe additive maps satisfying for all in the following settings: where is the prime field of These maps are so-called commuting on 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 if and only if G is commuting on Our investigation will lead us to consider functional identities on of the form with r, s belonging to a 2-point subset of At the end, we will discuss about the existence of non-standard additive commuting maps on which are not commuting on the set of rank 1 matrices and vice versa.