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

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 ⊆ Cα,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.