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Let A be a unital prime *-algebra (over the complex field C ) with a nontrivial projection. For any S 1 , S 2 , ... , S n ∈ A , define q 1 (S 1) = S 1 , q 2 (S 1 , S 2) = [ S 1 , S 2 ] ⋄ = S 1 S 2 − S 2 S 1 ∗ and q n (S 1 , S 2 , ... , S n) = [ q n − 1 (S 1 , S 2 , ... , S n − 1) , S n ] ⋄ for all integers n ≥ 3. In this article, it is shown that if a map ξ : A → A (not necessarily linear) satisfies ξ (q n (S 1 , S 2 , ... , S n)) = ∑ i = 1 n q n (S 1 , ... , S i − 1 , ξ (S i) , S i + 1 , ... , S n) (n ≥ 3) for all S 1 , S 2 , ... , S n ∈ A with S 1 ∗ S 2 ∗ ⋯ S n ∗ = 0 , then ξ is an additive *-derivation. Finally, this result is applied to standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

Assume that is a unital algebra over a commutative unital ring ℛ and is an -bimodule. A trivial extension algebra ⋉ is defined as an ℛ -algebra with usual operations of ℛ -module × and the multiplication defined by (u 1 , s 1) (u 2 , s 2) = (u 1 u 2 , u 1 s 2 + s 1 u 2) for all u 1 , u 2 ∈ , s 1 , s 2 ∈. In this paper, we prove that under certain conditions every Jordan n -derivation Δ on ⋉ can be expressed as Δ = d + δ , where d is a derivation and δ is both a singular Jordan derivation and an antiderivation. As applications, we characterize Jordan n -derivations on triangular algebras and generalized matrix algebras. [ABSTRACT FROM AUTHOR]

Let 프 be a triangular ring and let p n ⁢ (U 1 , U 2 , ... , U n) denote the (n - 1) th commutator of elements U 1 , U 2 , ... , U n ∈ 프 . Suppose that ℕ is the set of nonnegative integers and 픏 = { ξ r } r ∈ ℕ is a sequence of additive mappings on 프 such that ξ 0 = i ⁢ d 프 , the identity mapping on 프 , and for each r ∈ ℕ , ξ r ⁢ (p n ⁢ (U 1 , U 2 , ... , U n)) = ∑ i 1 + i 2 + ⋯ + i n = r p n ⁢ (ξ i 1 ⁢ (U 1) , ξ i 2 ⁢ (U 2) , ... , ξ i n ⁢ (U n)) for all U 1 , U 2 , ... , U n ∈ 프 with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 . In this paper, it is shown that under certain conditions 픏 = { ξ r } r ∈ ℕ has the standard form, that is, there exist a higher derivation { d r } r ∈ ℕ on 프 and a family { h r } r ∈ ℕ of additive mappings h r : 프 → 풵 ⁢ (프) satisfying h r ⁢ (p n ⁢ (U 1 , U 2 , ... , U n)) = 0 for all U 1 , U 2 , ... , U n ∈ 프 with U 1 ⁢ U 2 ⁢ ⋯ ⁢ U n = 0 such that for each r ∈ ℕ , ξ r ⁢ (U) = d r ⁢ (U) + h r ⁢ (U) for all U ∈ 프 . [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity, A, B be R-algebras and M be an (A, B)-bimodule. Let T = Tri(A, M, B) be a (n -- 1)-torsion free triangular algebra. In this article, we prove that every multiplicative Lie n-higher derivation on triangular algebras has the standard form. Also, the main result is applied to some classical examples of triangular algebras such as upper triangular matrix algebras and nest algebras. [ABSTRACT FROM AUTHOR]