GENERALIZED SKEW DERIVATIONS AS JORDAN HOMOMORPHISMS ON MULTILINEAR POLYNOMIALS

Let R be a prime ring of characteristic different from 2, Qr be its right Martindale quotient ring and C be its extended cen- troid. Suppose that G is a nonzero generalized skew derivation of R, � is the associated automorphism of G, f(x1,...,xn) is a non-central multilinear polynomial over C with n non-commuting variables and S = {f(r1,...,rn)|r1,...,rn 2 R}. If G acts as a Jordan homomorphism on S, then either G(x) = x for all x 2 R, or G = �. In all that follows let R be a prime ring, Z(R) the center of R, Qr be the right Martindale quotient ring of R and C = Z(Qr) be the center of Qr. C is usually called the extended centroid of R and is a field when R is a prime ring. It should be remarked that Qr is a centrally closed prime C-algebra. We refer the reader to (6) for the definitions and the related properties of these objects. We recall that an additive map d on R is called a derivation if d(xy) = d(x)y +xd(y) for all x,y ∈ R. Starting from this definition we may introduce another concept of an additive function which generalizes derivations: the addi- tive map G of R is said to be a generalized derivation if G(xy) = G(x)y +xd(y) for all x,y ∈ R, where d is a derivation of R (usually G is said to be a gener- alized derivation associated with d). Obviously, any derivation of R and any map of R with form f(x) = ax +xb for some a,b ∈ R, are both generalized derivations. The latter are usually called inner generalized derivations. We would like to point out that one of the leading roles in the development of the theory of generalized derivations is played by the inner generalized derivations. We say that an additive map F acts as a homomorphism on a subset T ⊆ R, if F(xy) = F(x)F(y) for all x,y ∈ T ; F acts as an anti-homomorphism on T , if F(xy) = F(y)F(x) for all x,y ∈ T ; finally F acts as a Jordan homomorphism on T if F(x 2 ) = F(x) 2 for all x ∈ T. Obviously any additive map, which is a homomorphism or an anti-homomorphism, is a Jordan homomorphism. On