Ideals and symmetric reverse bi-derivations of prime and semiprime rings

Abstract

Let $R$ be a prime ring of char $R \neq 2$ and $I$ a nonzero ideal of $R$. Suppose that there exist symmetric reverse bi-derivations $D_1(.,):. R X R \rightarrow R$ and $D_2(.,):. R X R \rightarrow R$ such that $D_1\left(d_2(x), x\right)=0$ for all $x \in I$, where $d_2$ denotes the trace of $D_2$. Then either $D_1=0$ or $D_2=0$.