Abstract:
The major objective of this project work was to discuss the proofs of some theorems that deal with ideals and commutativity of prime rings with generalized derivations. In algebra, several methods were used to show prime rings are commutative rings under a certain given conditions. This project investigated the results that asserted the existence of a generalized derivation on ideals and prime rings which forces the prime ring to be commutative. We discussed the proofs of some theorems on one sided ideals and commuting conditions on generalized derivations in prime rings which facilitates prime ring to be commutative. Some results of generalized derivation on prime rings satisfying certain differential identities have been also discussed. Illustrative examples were presented to clarify more the concepts and to show the necessity of derivation and generalized derivation of prime rings in the proved. Further, different lemmas and some of the related concepts were used in the proof of the theorems that show a commutativity of prime rings with generalized derivation in the setting of ideals.
