Journal Details Format
Journal
First Published
16 Jun 2010
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2 times per year
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# (ϕ, φ)-derivations on semiprime rings and Banach algebras

###### Accepted: 04 Dec 2019
Journal Details Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English

Let 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, Gscr; : ℛ ℛ satisfying the relations 2𝒟(xn)=𝒟(xn-1)φ(x)+ϕ(xn-1)𝒢(x)+𝒢(x)φ(xn-1)+ϕ(x)𝒢(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{G}\left( x \right) + \mathcal{G}\left( x \right)\phi \left( {{x^{n - 1}}} \right) + \varphi \left( x \right)\mathcal{G}\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.

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• #### A note on the volume of ∇-Einstein manifolds with skew-torsion

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