Afr. Mat. (2013) 24:503–510 DOI 10.1007/s13370-012-0075-9 Lie ideals and generalized (α, β)-derivations of ∗-prime rings Nadeem ur Rehman · Radwan Mohammed AL-Omary · Shuliang Huang Received: 21 October 2011 / Accepted: 21 March 2012 / Published online: 17 April 2012 © African Mathematical Union and Springer-Verlag 2012 Abstract Let ( R, ∗) be a 2-torsion free ∗-prime ring with involution ∗, L  = 0 be a square closed ∗-Lie ideal of R and α, β automorphisms of R commuting with ∗. An additive mapping F : R → R is called a generalized (α, β)-derivation on R if there exists an (α, β)-derivation d such that F (xy ) = F (x )α( y ) + β(x )d ( y ) holds for all x , y ∈ R. In the present paper, we shall show that L ⊆ Z ( R) such that R is a ∗-prime ring admits a generalized (α, β)- derivation satisfying several conditions, but associated with an (α, β)-derivation commuting with ∗. Keywords ∗-ideals ·∗-prime rings · Derivations and generalized (α, β)-derivations Mathematics Subject Classification 16D90 · 16W25 · 16N60 · 16U80 1 Introduction Let R be an associative ring with center Z ( R) and involution ∗. For each x , y ∈ R, the symbol [x , y ] will represent the commutator xy - yx and the symbol x ◦ y stands for the skew-commutator xy + yx . An additive mapping x  → x ∗ on a ring R is called an involution if ((x ) ∗ ) ∗ = x and (xy ) ∗ = y ∗ x ∗ . A left (resp. right, two sided) ideal L of R is called a left (resp. right, two sided) ∗-ideal if L ∗ = L . An ideal P of R is called a ∗-prime ideal if P ( = R) N. Rehman’s research is supported by UGC, India, Grant No. 36-8/2008(SR). N. Rehman (B ) Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India e-mail: rehman100@gmail.com R. M. AL-Omary Department of Mathematics, Al-Naderah Faculty, Ibb University, Ibb, Yemen S. Huang Department of Mathematics, Chuzhou University, Chuzhou, 239012 Anhui, People’s Republic of China 123