scholarly journals THE JORDAN DERIVATIONS OF SEMIPRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS

2016 ◽  
Vol 29 (4) ◽  
pp. 531-542
Author(s):  
Byung-Do Kim
Keyword(s):  
Banach Algebras ◽  
Semiprime Rings

scholarly journals A note on a pair of derivations of semiprime rings

2004 ◽  
Vol 2004 (39) ◽  
pp. 2097-2102 ◽  
Author(s):  
Muhammad Anwar Chaudhry ◽  
A. B. Thaheem
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Semiprime Rings

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: letRbe a semiprime ring with centerZ(R), and letf,gbe derivations ofRsuch thatf(x)x+xg(x)∈Z(R)for allx∈R, thenfandgare central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivationfof a semiprime ringRis skew-commuting.

scholarly journals A pair of generalized derivations in prime, semiprime rings and in Banach algebras

2021 ◽  
Vol 39 (4) ◽  
pp. 131-141
Author(s):  
Basudeb Dhara ◽  
Venus Rahmani ◽  
Shervin Sahebi
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Banach Algebras ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Fixed Integer ◽  
Semiprime Rings ◽  
Commutative Banach Algebras

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy)+G(yx))n= (xy ±yx) for all x,y ∈ I, then one ofthe following holds:(1) R is commutative;(2) n = 1 and H(x) = x and G(x) = ±x for all x ∈ R.Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

Some differential identities on prime and semiprime rings and Banach algebras

2018 ◽  
Vol 68 (2) ◽  
pp. 305-313 ◽  
Author(s):  
Mohd Arif Raza ◽  
Mohammad Shadab Khan ◽  
Nadeem ur Rehman
Keyword(s):  
Banach Algebras ◽  
Differential Identities ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

On derivations in prime and semiprime rings with Banach algebras

2018 ◽  
Vol 12 (8) ◽  
pp. 297-309
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeem ur Rehman
Keyword(s):  
Banach Algebras ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings

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

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bilal Ahmad Wani
Keyword(s):  
Semiprime Ring ◽  
Banach Algebras ◽  
Fixed Integer ◽  
Semiprime Rings ◽  
Additive Mappings

Abstract 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 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 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 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒢 ( x ) + 𝒢 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒢 ( x n - 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), 2 𝒢 ( x n ) = 𝒢 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) D ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 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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