Algebraic Identities on Generalized Derivations in Prime Rings with Involution
Authors: M. El hamdaoui  
Published: November 2024

This research paper aims to investigate the commutativity properties of prime rings in relation to generalized derivations and a left multiplier that fulfil specific algebraic conditions involving involution. Additionally, we present various examples studies to illustrate that the constraints imposed in our theorems are indeed necessary and cannot be omitted without compromising the validity of our results.Introduction: Consider a ring S that satisfies the associative property, and let Z(S) represent its center. An involution denoted by *, which is an additive function mapping S to itself. This involution has specific properties: for any α and β in S, applying the involution twice returns the original element ((α^* )^*=α), it distributes over addition ((α+β)^*=α^*+β^*), and it reverses the order of multiplication ((αβ)^*=β^* α^*). We categorize elements as hermitian when they remain unchanged under the involution (α^*=α), and as skew-hermitian when they change sign (α^*=-α). We use H(S) to represent the collection of all hermitian elements in S, and S(S) for all skew-hermitian elements. The involution is classified as first kind if H(S) is a subset of the center of S, denoted as Z(S). If this is not the case, it’s considered second kind, and in this scenario, the intersection of H(S) and Z(S) contains more than just the zero element. We also define several types of mappings on ring S. A left multiplier, △, is an additive map where △(υω)=△(υ)ω, ∀υ,ω∈S. A derivation, ψ, is an additive mapping that satisfies ψ(υω)=ψ(υ)ω+υψ(ω),  ∀υ,ω∈S. Extending this concept, we define a generalized derivation, Γ, which is linked to a derivation ψ. This function satisfies Γ(υω)=Γ(υ)ω+υψ(ω),  ∀υ,ω∈S. It’s worth noting that any derivation can be considered a generalized derivation associated with itself.Objectives: In this study, we intend to examine the commutativity of a prime ring S by utilizing generalized derivations Γ_1, Γ_2, and a left multiplier △, while adhering to specific algebraic identities that involve involution. Specifically, we will delve into the commutativity of rings S that fulfill the following algebraic conditions:      •  [Γ_1 (υ),Γ_2 (υ^*)]+△([υ,υ^*])∈Z(S), for all υ∈S;     •  Γ_1 (υ)∘Γ_2 (υ^*)+△(υ∘υ^*)∈Z(S), for all υ∈S;     •  [Γ_1 (υ),Γ_2 (υ^*)]+△(υ∘υ^*)∈Z(S), for all υ∈S;     •  Γ_1 (υ)∘Γ_2 (υ^*)+△([υ,υ^*])∈Z(S), for all υ∈S;     •  Γ(υυ^*)±Γ(υ)Γ(υ^*)+△([υ,υ^*])∈Z(S), for all υ∈S;     •  Γ(υυ^*)±Γ(υ^*)Γ(υ)+△(υ∘υ^*)∈Z(S), for all υ∈S. Lastly, we offer examples to illustrate that the constraints applied to our hypotheses are necessary and not redundant.Results: Building on the work of Nejjar ([8], Theorems 3.5, 3.8), who demonstrated that a 2-torsion-free prime ring with involution and a derivation ψ satisfying certain conditions must be commutative, we explore broader generalizations of these conditions. Specifically, Nejjar showed that if the derivation ψ meets either of the following criteria: [ψ(υ),ψ(υ)]±υ∘υ∈Z(S),  ∀υ∈S or ψ(υ)∘ψ(υ)±υ∘υ∈Z(S),  ∀υ∈S, then S is necessarily commutative. Our work extends these findings by introducing new identities for pairs of generalized derivations that are connected to a left multiplier △. Finally, we offer examples to illustrate that the constraints applied to our hypotheses are necessary and not redundant.Conclusions: In this research, we investigate the commutativity of prime rings S admitting an involution and generalized derivations satisfying some algebraic identities. We can conclude our paper with an open question.Open question: are these results correct if we replace the generalized derivation by the generalized (α,β)-derivation, where α and β are automorphisms of ring S?