On functional identities and d-free subsets of rings.: I

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 a:S -> A be a map, let m be a positive integer and let E-i, F-i: Sm-1 -> Q, 1 less than or equal to i less than or equal to m, be maps. We study functional identities of the form Sigma(i=1)(m) (E-i(i) s(i)(alpha) + s(i)(alpha) F-i(i)) epsilon C for all s(1),s(2),...,s(m) epsilon S (where E-i(i) means E-i(s(1),...;<(s(i))over cap>,.... s(m)), etc). If S-alpha is an (nt + 1)-free subset of Q (for example, if S-alpha = A and A is not algebraic of bounded degree m + 1 over C, or S-alpha is a noncentral Lie ideal of A and A is not algebraic of bounded degree m + 2 over C), definitive results are obtained.