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This paper represents the structures of generalized derivations satisfying an identity with n -Engel condition. To prove our result, we take R as a prime ring with char (R) ≠ 2 , and U as Utumi quotient ring of R. The center of U is C which is called extended centroid of R. Let L be a non-central Lie ideal of R. Suppose that F and G are two nonzero generalized derivations of R and n ≥ 1 , any fixed integer. If [ F (u) G (u) , u ] n = 0 for all u ∈ L , then one of the below-mentioned conclusions holds: (1) F (x) = x a and G (x) = b x for all x ∈ R and for some a , b ∈ U with a b ∈ C ; (2) R satisfies s 4 . [ABSTRACT FROM AUTHOR]

Let R be a prime or semiprime ring, λ a nonzero left-sided ideal of R and α , β be two automorphisms of R. Let F : R → R and G : R → R be two generalized (α , β) -derivations of R associated with (α , β) -derivations d : R → R and g : R → R , respectively. The purpose of this paper is to investigate the following identities: (1) G (x y) + d (x) F (y) + α (x y) = 0 ; (2) G (x y) + d (x) F (y) = 0 ; (3) G (x y) + d (x) F (y) + α (y x) = 0 ; (4) G (x y) + d (x) F (y) + α (y x) + α (x y) = 0 ; (5) G (x y) + d (x) F (y) + α (y x) − α (x y) = 0 ; (6) G (x y) + d (y) F (x) + α (y x) = 0 ; (7) G (x y) + d (y) F (x) + α (y x) + α (x y) = 0 ; (8) G (x y) + d (y) F (x) + α (y x) − α (x y) = 0 ; (9) G (x y) + F (x) F (y) = 0 ; (10) G (x y) + F (y) F (x) = 0 ; (11) F (x y) + G (x) α (y) + α (y x) = 0 ; (12) F (x) F (y) + G (x) α (y) + α (y x) = 0 , for all x , y ∈ λ. [ABSTRACT FROM AUTHOR]

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]

Let R be an associative ring and σ, τ two epimorphisms of R. A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)-(σ, τ)-derivation if F(xy) = F(x)σ(y) + τ(x)g(y) holds for all x, y ∈ R, where g is any mapping on R. The main object in this article is to study the situations (1) F(xy) - F(x)F(y) = 0, (2) F(xy) + F(x)F(y) = 0, (3) F(xy) - F(y)F(x) = 0 and (4) F(xy) + F(y)F(x) = 0; for all x, y in some nonzero subsets of prime and semiprime rings, where F is a multiplicative (generalized)-(σ, σ)-derivation of R. [ABSTRACT FROM AUTHOR]