A Note on Multiplicative (Generalized) - (α, β) - Reverse Derivations on Left Ideals in Prime Rings

Abstract

A mapping G: R→ R (not necessarily additive) is called multiplicative right αcentralizer if T(xy) = α(x)T(y) for all x, y ∈ R. A mapping G: R → R (not necessarily additive) is called multiplicative (generalized) -(α, β) -reverse derivation if there exists a map (neither necessarily additive or derivation) g : R→ R such that G(xy) = G(y)α(x) + β(y)g(x) for all x, y ∈ R, where α and β are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized)-(α, β)-reverse derivations and multiplicative right α-centralizer on the left ideal of a prime ring R. The main objective of the present paper is to investigate the following algebraic identities:and (v) G(xy) ± T(x)G(y) = 0 for all x, y in an appropriate subset of R.

Multiplicative (Generalized)-(Σ,τ)-Reverse Derivations in Prime Rings

“…In 2019, Nadeem ur Rehman et.al [10] proved some results on a note on multiplicative (generalized)-(𝛼, 𝛽)-reverse derivations in prime rings. In 2022, Jaya Subba Reddy et.al [9] proved some results on a note on multiplicative (generalized)-(𝛼, 𝛽)-reverse derivations on left ideals in prime rings. In this paper, we proved some results on multiplicative (generalized) (𝜎, 𝜏)-reverse derivations in prime rings.…”