On functional identities and d-free subsets of rings, I

Abstract

Let A be a prime ring with maximal right ring of quotients Q and with extended centroid C. Further, let S be a set, let α : S → A be a map, let m be a positive integer and let Ei, Fi : Sm-1 → Q, 1 ≤ i ≤ m, be maps. We study functional identities of the form Σmi=1 (Eiisαi + sαiFii) ∈ C for all s1, s2, . . . , sm ∈ S (where Eii means Ei(s1, . . . , si, . . . . sm), etc). If Sα is an (m + 1)-free subset of Q (for example, if Sα = A and A is not algebraic of bounded degree m + 1 over C, or Sα is a noncentral Lie ideal of A and A is not algebraic of bounded degree m + 2 over C), definitive results are obtained.