UPSI Digital Repository (UDRep)
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UPSI Digital Repository (UDRep)
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| Abstract : Perpustakaan Tuanku Bainun |
| The concept of derivation on rings is an old problem and plays a very important role in algebra,
analysis, geometry and physics. The concept of derivation was generalized in different directions
as centralizing derivation, Lie and Jordan derivations, (θ, φ)- derivation, generalized derivations
and reverse derivations. The main aim of this research is to generalize these concepts. Therefore,
the action of (θ, φ)- derivations on Lie and Jordan Ideals of prime rings with characteristic not
equal two will be investigated. The next objective was to determine the commutativity of a prime
ring when the generalized derivation acting on the Jordan ideal of a prime ring. Furthermore, this
research will extend the centralizing derivations and reverse derivation on δ- ideal for δ- prime
rings. Finally, we extended the concepts of generalized reverse derivation and generalized reverse
(θ, φ)- derivation on a δ- prime and δ- semiprime rings. Some techniques and methods of the theory
of commutative rings will be significantly used in this research, such as the properties of
commutators and centralizing derivations. The findings of the research showed that the prime ring,
R is commutative if there is a generalized derivation with a specific condition on R. Also, some
results depending on a generalized (θ, φ)- derivation on Jordan ideal of R are extended. Beside,
the relationship between some kinds of derivations as centralizing δ- derivations and the
commutativity of δ- prime rings was developed. Finally, the commutativity of δ- prime ring with
centralizing reverse δ- derivation was obtained. It can be concluded that the study of derivation
and reverse derivation on a prime and δ- prime rings have considered some conditions to obtain the
proven results. As an implication, the obtained results in this research can be used as a guide for
other researchers to extend new results related to semiprime rings in the future.
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