Notes on symmetric bi-(α,α)-derivations in rings

Abstract

Let R be a prime ring with center Z, I a nonzero ideal of R and D: R × R → R a symmetric bi–(α, α)-derivation and d be the trace of D. In the present paper, we have considered the following conditions: i) [d(x), x]_α,α = 0, ii)[d(x), x]α,α ⊆ C_α,α, iii)(d(x), x)_α,α = 0, iv)D₁(d₂(x), x) = 0, v)d₁(d₂(x)) = f(x), for all x, y ∈ I, where D₁ and D₂ are two symmetric bi-(α, α)-derivations, d₁, d₂ are the traces of D₁, D₂ respectively, B: R × R → R is a symmetric bi-additive mapping, f is the trace of B.