On generalized homoderivations of prime rings - Rehman, N.; Sogutcu, E. K.; Alnoghashi, H. M. (Matematychni Studii, 2023)
Let A be a ring with its center Z(A). An additive mapping ξA→ A is called a homoderivation on A if a,bAξ(ab)=ξ(a)ξ(b)+ξ(a)b+aξ(b). An additive map ψA→ A is called a generalized homoderivation with associated homoderivation ξ on A if a,bAψ(ab)=ψ(a)ψ(b)+ψ(a)b+aξ(b). This study examines whether a prime ring A with a generalized homoderivation ψ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities: ψ(a)ψ(b)+abZ(A), ψ(a)ψ(b)-abZ(A), ψ(a)ψ(b)+abZ(A), ψ(a)ψ(b)-abZ(A), ψ(ab)+abZ(A), ψ(ab)-abZ(A), ψ(ab)+baZ(A), ψ(ab)-baZ(A)(a, bA). Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous.