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

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