COMMUTATIVITY OF GAMMA RINGS WITH GENERALIZED DERIVATIONS IN THE SETTING OF IDEALS SATISFYING CERTAIN DIFFERENTIAL IDENTITIES

Abstract

The main purpose of this project is to elaborate the ideas and concepts of commutativity of gamma rings with generalized derivations in the setting of ideals satisfying certain differential identities. The important preliminary concepts, definitions, lemmas and theorems were elaborated with detailed explanatory steps to make the concept clear. In addition, to study commutativity of gamma rings definitions and properties of generalized derivation and nonzero ideals have been employed. Detailed explanatory steps on the proofs of the theorems and lemmas on commutativity of gamma rings admitting a generalized derivation satisfying certain differential identities were provided. It is discussed the detailed proofs on commutativity of gamma rings in a prime gamma rings which admits a nonzero generalized derivation contained in the nonzero ideals of 𝑍(𝑀). Also, It discussed some differential identities of a prime gamma ring that admits a nonzero generalized derivation associated with nonzero derivations on ideal I which have been commutative gamma rings. Moreover, we identified the sufficient conditions on which the assumptions is a 2-torsion free prime gamma rings and is a nonzero ideal I of M that admits a nonzero derivation d have been commutative gamma ring that a prime gamma ring 𝑀 is commutative if 𝐷 is a generalized derivation on 𝑀 with an associated non-zero derivation 𝑑 on 𝑀 such that 𝐼 is centralizing and commuting on a left ideal 𝐼 of 𝑀.Commutativity of gamma rings with generalized derivations in the setting of ideals was presented and showed that a gamma ring 𝑀 with a generalized derivation admitting a non zero derivation such that a nonzero ideal 𝐼 of 𝑀 satisfyingthe foollowing differential identities (i) 𝐷(𝑥)𝛾𝐷(𝑦)=[𝑥,𝑦]𝛾 (ii) 𝐷(𝑦)𝛾𝐷(𝑥)=[𝑥,𝑦]𝛾 (iii) 𝐷(𝑦)𝛾𝐷(𝑥)=[𝑥𝑜𝛾𝑦] for all 𝑥,𝑦 ∈𝐼;𝛾∈ Γ be a commutative gamma rings.. Finally, we have seen that the sufficient conditions on which the assumptions is a nonzero ideals of prime gamma ring generalized derivation acts as a right derivation or a left centralizer associated with a nonzero derivation is commutative gamma ring