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Let 풜 be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ which is closed under adjoint operation. Suppose that ϕ : 풜 → 풜 is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ -derivation. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with A ⁢ (R) its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by AG ¯ ⁢ (R) , is an undirected graph with vertex set A ⁢ (R) * = A ⁢ (R) ∖ { 0 } and two vertices I 1 and I 2 are adjacent if and only if I 1 m ⁢ I 2 n = 0 with I 1 m ≠ 0 and I 2 n ≠ 0 , for some positive integers m and n. In this paper, we first study some basic properties of AG ¯ ⁢ (R) and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ (R) , the annihilator-ideal graph A I ⁢ (R) and the annihilating-ideal graph AG ⁢ (R) of a commutative ring R. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation (ℱ , ξ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential. [ABSTRACT FROM AUTHOR]