Some results concerning symmetric generalized skew biderivations on prime rings

Let R be a ring. A biadditive symmetric mapping D : R × R −→ R is called a symmetric skew biderivation if for every x ∈ R, the map y → D(x, y) is a skew derivation of R (as well as for every y ∈ R, the map x → D(x, y) is a skew derivation of R). Let D : R×R −→ R be a symmetric biderivation. A biadditive symmetric mapping ∆ : R × R −→ R is said to be a symmetric generalized skew biderivation if for every x ∈ R, the map y → ∆(x,y) is a generalized skew derivation of R associated with D (as well as for every y ∈ R, the map x → ∆(x, y) is a generalized skew derivation of R associated with D). In this paper we study some commutativity conditions for a prime ring R related to the behaviour of the trace of symmetric generalized skew biderivations of R.