On certain functional equation related to derivations

Abstract

In this article, we prove the following result. Let n ≥ 3 n\ge 3 be some fixed integer and let R R be a prime ring with char ( R ) ≠ ( n + 1 ) ! 2 n − 2 {\rm{char}}\left(R)\ne \left(n+1)\!{2}^{n-2} . Suppose there exists an additive mapping D : R → R D:R\to R satisfying the relation 2 n − 2 D ( x n ) = ∑ i = 0 n − 2 n − 2 i x i D ( x 2 ) x n − 2 − i + ( 2 n − 2 − 1 ) ( D ( x ) x n − 1 + x n − 1 D ( x ) ) + ∑ i = 1 n − 2 ∑ k = 2 i ( 2 k − 1 − 1 ) n − k − 2 i − k + ∑ k = 2 n − 1 − i ( 2 k − 1 − 1 ) n − k − 2 n − i − k − 1 x i D ( x ) x n − 1 − i \begin{array}{rcl}{2}^{n-2}D\left({x}^{n})& =& \left(\mathop{\displaystyle \sum }\limits_{i=0}^{n-2}\left(\genfrac{}{}{0.0pt}{}{n-2}{i}\right){x}^{i}D\left({x}^{2}){x}^{n-2-i}\right)+\left({2}^{n-2}-1)\left(D\left(x){x}^{n-1}+{x}^{n-1}D\left(x))\\ & & +\mathop{\displaystyle \sum }\limits_{i=1}^{n-2}\left(\mathop{\displaystyle \sum }\limits_{k=2}^{i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{i-k}\right)+\mathop{\displaystyle \sum }\limits_{k=2}^{n-1-i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{n-i-k-1}\right)\right){x}^{i}D\left(x){x}^{n-1-i}\end{array} for all x ∈ R . x\in R. In this case, D D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring with char ( R ) ≠ 2 {\rm{char}}\left(R)\ne 2 is a derivation.