On compositions of derivations of prime rings

Abstract:

Let $R$ be a prime ring and $\varphi \left ( {{x_i}} \right )$ be a differential polynomial of $R$. It is shown that if $\varphi \left ( {{x_i}} \right ) = 0$ holds on a nonzero two-sided ideal of $R$, then $\varphi \left ( {{x_i}} \right ) = 0$ holds on ${R_F}$, the left Martindale quotient ring of $R$. Using this together with Kharchenko’s theorem on differential identities, we settle three problems raised by Krempa and Matczuk in the positive.