Annihilating and Engel conditions on right ideals with generalized derivations

Abstract

Let $R$ be a prime ring of characteristic different from $2$, $U$ its right Utumi quotient ring, $C$ its extended centroid, $G$ a non-zero generalized derivation of $R$, $a\ne 0$ be an element of $R$, $I$ a non-zero right ideal of $R$ such that $s_4(I,\ldots ,I)I\ne 0$ and $n,k\ge 1$ fixed integers. If $a[G([r_1,r_2]^n),[r_1,r_2]^n]_k=0$, for any $r_1,r_2 \in I$, then either there exist $c \in U$ and $\gamma \in C$, such that $G(x)=cx$ and $(c-\gamma )I=0$, or $aI=aG(I)=0$.