Some Results on Left-sided Ideals of Semiprime Rings with Symmetric \((\alpha ,\beta )\) n-Derivations

Let R be a prime or semiprime ring and \(\alpha ,\beta \) are automorphisms of R. An n-additive mapping \(\Delta : R^n \rightarrow R\) is said to be \((\alpha ,\beta )\) n-derivation if \(\Delta (h_1,h_2,...,\) \(x_ix_i',...,h_n) = \Delta (h_1,h_2,...,x_i,...,h_n)\alpha (x_i') + \beta (x_i)\Delta (h_1,h_2,...,x_i',...,h_n)\) . In the present paper, we shall prove that the map \(\Delta :R \times R \times ...\times R \rightarrow R\) is zero if it satisfies the identity \([\delta (x), \beta (x)] = 0\) and \(\delta (x) \circ \beta (x)= 0\) for all \(x \in I\) , where I is a left ideal of R and \(\delta \) be the trace of \(\Delta \) . This result is also the generalization of Fošner result [7, Theorem 1].