关键词: prime rings, (α, β)-derivations and generalized (α, β)-derivations, Lie ideals, Morita context

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