On (?,?)-Derivations and Commutativity of Prime and Semi prime ?-rings Afrah Mohammad Ibraheem

الملخص

Let R be a ?-ring, and ?, ? be two automorphisms of R. An additive mapping d from a ?-ring R into itself is called a (?,?)-derivation on R if d(a?b) = d(a)? ?(b) + ?(a)?d(b), holds for all a,b ?R and ???. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)]? = [a,b]_?^((?,?) ) holds for all a,b?R and ???. In this paper, we investigate the commutativity of R by the strong commutativity preserving (?,?)-derivation d satisfied some properties, when R is prime and semi prime ?-ring.