Radwan Mohammed Al-Omary, Syed Khalid Nauman Ibb University; Jinnah University for Women
Abstract : Let $R$ be a ring with characteristic different from 2. An additive mapping $F:R\rightarrow R$ is called a generalized derivation on $R$ if there exists a derivation $d:R\rightarrow R$ such that $F(xy)=F(x)y+xd(y)$ holds for all $x,y\in R$. In the present paper, we show that if $R$ is a prime ring satisfying certain identities involving a generalized derivation $F$ associated with a derivation $d$, then $R$ becomes commutative and in some cases $d$ comes out to be zero (i.e., $F$ becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.
Keywords : Prime rings, derivations and generalized derivations, left centralizer