Lie ideals, Morita context and Generalized (α, β)-Derivations

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 techniques 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.