Left (Right) Centralizer of σ--Square Closed Lie Ideals of σ--Prime Rings

Let R be a σ-prime ring and F be a nonzero left (right) centralizer of R. This work includes two parts. In the first part, when I is a nonzero σ-ideal of R we prove that (i) if F commutes with σ on I and [x, R]IF (x) = (0) for all x ∈ I, then R is commutative. (ii) If r ∈ Sa sub σ /sub (R) or F commutes with σ on I and [F (x), r] = 0 for all x ∈ I, then r ∈ Z(R). (iii) If r ∈ Sa sub σ /sub (R) such that F ([x, r]) = 0 for all x ∈ R, then r ∈ Z(R). (iv) If R is a 2-torsion free σ-prime ring and F ([x, y]) = 0 for all x, y ∈ R, then R is a commutative ring. In the second part, when R is a 2-torsion free and U is a nonzero σ-square closed Lie ideal of R such that U cf: Z(R) we prove that: (i) if r ∈ U ∩ Sa sub σ /sub (R) and [F (x), r] = 0 for all x ∈ U , then r ∈ Z(R). (ii) If r ∈ U ∩ Sa sub σ /sub (R) and F ([x, r]) = 0 for all x ∈ U , then r ∈ Z(R).