Gen. Math. Notes, Vol. 33, No. 2, April 2016, pp.72-77 ISSN 2219-7184; Copyright c ICSRS Publication, 2016 www.i-csrs.org Available free online at http://www.geman.in Left (Right) Centralizer of σ −Square Closed Lie Ideals of σ −Prime Rings Ay¸ se Mutlu 1 and Ne¸ set Aydın 2 1,2 Department of Mathematics C ¸anakkale Onsekiz Mart University, C ¸anakkale, Turkey 1 E-mail: mutluaysee@gmail.com 2 E-mail: neseta@comu.edu.tr (Received: 4-3-16 / Accepted: 10-4-16) Abstract 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 σ (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 σ (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  Z (R) we prove that: (i) if r ∈ U ∩ Sa σ (R) and [F (x),r]=0 for all x ∈ U , then r ∈ Z (R). (ii) If r ∈ U ∩ Sa σ (R) and F ([x, r]) = 0 for all x ∈ U , then r ∈ Z (R). Keywords: σ−prime ring, σ−ideal, centralizer. 1 Introduction Throughout this paper, R will represent an associative ring with center Z (R). R is said to be 2−torsion free if whenever 2x =0, then x =0. An additive mapping σ : R → R is called an involution if σ is an anti-homomorphism and σ(σ(x)) = x for all x ∈ R. R is called σ−prime ring where σ is an involution of R if aRb = aRσ(b) = (0) implies that a = 0 or b = 0. A nonempty subset A of R is called σ−invariant if σ (A) ⊆ A. An ideal I of R is a σ−ideal if I