Acta Mathematica Scientia 2013,33B(4):1059–1070 http://actams.wipm.ac.cn LIE IDEALS, MORITA CONTEXT AND GENERALIZED (α, β)-DERIVATIONS ∗ S. Khalid NAUMAN Department of Mathematics, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia E-mail: snauman@kau.edu.sa Nadeem ur REHMAN Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India E-mail: rehman100@gmail.com R. M. AL-OMARY Department of Mathematics, Ibb University, Ibb, Yemen E-mail: radwan959@yahoo.com Abstract A classical problem in ring theory is to study conditions under which a ring is forced to become commutative. Stimulated from Jacobson’s famous result, several tech- niques are developed to achieve this goal. In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative. Key words prime rings; (α, β)-derivations and generalized (α, β)-derivations; Lie ideals; Morita context 2010 MR Subject Classification 16W25; 16N60; 16U80; 16D90 1 Introduction In mid forties in [3] Jacobson proved that “for every element r in a ring R, if r n(r) = r, for some positive integer n(r), then R is commutative”. Inspired from this result, several techniques are developed to investigate conditions under which a ring becomes commutative, for instance, generalizing Herstein’s conditions, using restrictions on polynomials, introducing derivations and generalized derivations on rings, looking special properties for rings, etc. For more details and references see the review article [6]. One can also achieve this goal by comparing two rings and imposing conditions on them. Let us assume that rings R and S are ingredients of a Morita context. It was observed in [4] that if a Morita context is semi-projective, in the sense that the Morita map on S is epic, and if R is commutative and S is reduced, then S becomes * Received May 16, 2012; revised October 5, 2012.