On Centrally Extended Generalized Jordan *-derivations and Related Maps in Rings

Let R be a 2-torsion-free noncommutative prime ring with involution * with center, Z. In this article, we introduce the notion of a centrally extended generalized Jordan *-derivation. We show that if R admits a centrally extended generalized Jordan *-derivation Γ associated with a centrally extended Jordan *-derivation θ , and satisfies the condition [Γ(a),a] ∈ Z (resp., [Γ(a),a*] ∈ Z) for all a ∈ R, then either Γ(a) = λ a and θ (a) = λ (a−a*) (resp., Γ(a) = λ a*) for all a ∈ R, where λ belongs to the extended centroid of R, or R satisfies the standard identity S4. Furthermore, we present examples to highlight the significance of the conditions imposed in our findings.