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 \(aI=aG(I)=0\).