# (ϕ, φ)-derivations on semiprime rings and Banach algebras

(ϕ, φ)-derivations on semiprime rings and Banach algebras AbstractLet ℛ be a semiprime ring with unity e and ϕ, φ be automorphisms of ℛ. In this paper it is shown that if ℛ satisfies 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒟(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1)2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 is an (ϕ, φ)-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟, 𝒢 : ℛ → ℛ satisfying the relations 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒟(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1)2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right)2𝒢(xn)=𝒢(xn-1)φ(x)+ϕ(xn-1)D(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1),2\mathcal{G}\left( {{x^n}} \right) = \mathcal{G}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right), for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 and 𝒢 are (ϕ, φ)--derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

# (ϕ, φ)-derivations on semiprime rings and Banach algebras

, Volume 29 (3): 13 – Dec 1, 2021
13 pages

/lp/de-gruyter/derivations-on-semiprime-rings-and-banach-algebras-QXKBfmow8d
Publisher
de Gruyter
eISSN
2336-1298
DOI
10.2478/cm-2021-0013
Publisher site
See Article on Publisher Site

### Abstract

AbstractLet ℛ be a semiprime ring with unity e and ϕ, φ be automorphisms of ℛ. In this paper it is shown that if ℛ satisfies 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒟(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1)2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right) for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 is an (ϕ, φ)-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟, 𝒢 : ℛ → ℛ satisfying the relations 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒟(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1)2\mathcal{D}\left( {{x^n}} \right) = \mathcal{D}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right)2𝒢(xn)=𝒢(xn-1)φ(x)+ϕ(xn-1)D(x)+𝒟(x)φ(xn-1)+ϕ(x)𝒟(xn-1),2\mathcal{G}\left( {{x^n}} \right) = \mathcal{G}\left( {{x^{n - 1}}} \right)\phi \left( x \right) + \varphi \left( {{x^{n - 1}}} \right)\mathcal{D}\left( x \right) + \mathcal{D}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{D}\left( {{x^{n - 1}}} \right), for all x ∈ ℛ and some fixed integer n ≥ 2, then 𝒟 and 𝒢 are (ϕ, φ)--derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

### Journal

Communications in Mathematicsde Gruyter

Published: Dec 1, 2021

Keywords: Prime ring; semiprime ring; Banach algebra; Jordan derivation; ( ϕ , φ )-derivation; 16N60; 46J10; 16W25