Search

Let R be a prime ring of characteristic different from 2, Z(R) its center, Qr its right Martindale quotient ring, C its extended centroid, f (x 1 , ... , x n) a non-central multilinear polynomial over C, b ∈ Q r and F, G two b-generalized skew derivations of R. If there exists a ∈ R \ Z (R) such that [ a , F (f (r 1 , ... , r n)) f (r 1 , ... , r n) ] = f (r 1 , ... , r n) G (f (r 1 , ... , r n)) for any r 1 , ... , r n ∈ R , then f (x 1 , ... , x n) 2 is central valued on R and there exist p , q ∈ Q r such that F(x) = px, G(x) = xq for all x ∈ R with [ a , p ] = q . [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R, $$C=Z(U)$$ the extended centroid of R, and $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that $$F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots ,x_n))F(f(x_1,\ldots ,x_n))=0$$ for all $$x_1,\ldots ,x_n\in R$$ . Then $$f(x_1,\ldots ,x_n)^2$$ is central valued on R and one of the following holds: [ABSTRACT FROM AUTHOR]