Some Results on (,) – Lie Ideals in Prime rings

Abstract

Let R be a prime with characteristic not equal two, σ,τ : RR be two automorphisms of R. and d be a nonzero derivation of R commuting with σ,τ .It is proved that :
1) Assume U ba a(σ,τ)-left Lie ideal of R.
(a) If [U,U], C, and [U,U]=(0) ,then UZ(R).
(b) If [U,U], C, , then UZ(R) .
(c) If (v)+(v)Z(R) , for some vU ,then there exists a nonzero left ideal A of R and a nonzero right ideal B of R such that [R,A], U , [R,B],U but [R,A], Cσ, and [R,B] Cσ, .
(d) If ))0()(()0()(aUdorUad for Ra, then a=0 or (u)+(u)Z(R) , for all uU.
2) If U be a(σ,τ)-Lie ideal of R for ,))(()(,,,CUadorCaUdRa)(RZa, then a=0 or
. )(RZU
Also, in this paper we study some results when characteristic of R equal two and we show that the condition characteristic of R not equal two can not be excluded.