DOI: 10.2478/s12175-012-0021-1 Math. Slovaca 62 (2012), No. 3, 451–460 ON GENERALIZED (θ,φ)-DERIVATIONS IN SEMIPRIME RINGS WITH INVOLUTION Mohammad Ashraf — Nadeem-ur-Rehman — Shakir Ali* — Muzibur Rahman Mozumder (Communicated by Constantin Tsinakis ) ABSTRACT. The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime ∗-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F : R → R associated with a (θ, φ)-derivation d of R such that F (xx ∗ )= F (x)θ(x ∗ )+ φ(x)d(x ∗ ) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained. c 2012 Mathematical Institute Slovak Academy of Sciences 1. Introduction Throughout the discussion, unless otherwise mentioned, R will denote an associative ring having at least two elements with center Z (R). However, R may not have unity. For any x, y ∈ R, the symbol [x, y] (resp. (x ◦ y)) will denote the commutator xy − yx (resp. the anti-commutator xy + yx). Recall that R is prime if aRb = {0} implies that a = 0 or b = 0. A ring R is called semiprime if aRa = {0} with a ∈ R implies a =0. An additive mapping x → x ∗ satisfying (xy) ∗ = y ∗ x ∗ and (x ∗ ) ∗ = x for all x, y ∈ R, is called an involution on R. A ring equipped with an involution is called a ∗-ring or ring with involution. 2010 Mathematics Subject Classification: Primary 16W10; Secondary 16W25, 16N60. K e y w o r d s: semiprime ring, involution, derivation, (θ,φ)-derivation, generalized derivation, generalized (θ, φ)-derivation, generalized Jordan (θ,φ)-derivation. This research is partially supported by grants from DST (Grant No. SR/S4/MS:556/08), UGC (Grant No. 36-8/2008(SR)) and UGC (Grant No. 39-37/2010(SR)).