Bollettino dell’Unione Matematica Italiana https://doi.org/10.1007/s40574-018-0177-x Lie generalized derivations on trivial extension algebras D. Bennis 1 · H. R. Ebrahimi Vishki 2 · B. Fahid 1 · A. A. Khadem-Maboudi 3 · A. H. Mokhtari 4 Received: 15 June 2018 / Accepted: 1 October 2018 © Unione Matematica Italiana 2018 Abstract In this paper, we investigate the problem of describing the form of Lie generalized derivations on trivial extension algebras. We show, under some conditions, that every Lie generalized derivation on a trivial extension algebra is an sum of a generalized derivation and a cen- ter valued map which vanishes on all commutators. As an application we characterize Lie generalized derivation on a triangular algebra. Keywords Lie generalized derivation · Trivial extension algebra · Triangular algebra Mathematics Subject Classification 15A78 · 16W25 · 47B47 1 Introduction Throughout the paper R will denote a commutative ring with unity, A will be a unital R- algebra with center Z ( A) and M will be a unital A-bimodule. For a ∈ A and m ∈ M, we use B H. R. Ebrahimi Vishki vishki@um.ac.ir D. Bennis d.bennis@fsr.ac.ma; driss_bennis@hotmail.com B. Fahid fahid.brahim@yahoo.fr A. A. Khadem-Maboudi amaboudi@sbmu.ac.ir A. H. Mokhtari a.mokhtari@birjand.ac.ir 1 Department of Mathematics, Faculty of Sciences, CeReMAR Center, Mohammed V University, B.P. 1014 Rabat, Morocco 2 Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran 3 Department of Biostatistics, Faculty of Paramedical Sciences, Shahid Beheshti University of Medical Sciences, Tehran, Iran 4 Technical Faculty of Ferdows, University of Birjand, Birjand, Iran 123