부산대학교 도서관

In this article, we prove the following result. Let n≥3n\ge 3 be some fixed integer and let RR be a prime ring with char(R)≠(n+1)!2n−2{\rm{char}}\left(R)\ne \left(n+1)\!{2}^{n-2}. Suppose there exists an additive mapping D:R→RD:R\to R satisfying the relation 2n−2D(xn)=∑i=0n−2n−2ixiD(x2)xn−2−i+(2n−2−1)(D(x)xn−1+xn−1D(x))+∑i=1n−2∑k=2i(2k−1−1)n−k−2i−k+∑k=2n−1−i(2k−1−1)n−k−2n−i−k−1xiD(x)xn−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, DD 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.