SOME RESULTS ON LIE IDEALS WITH SYMMETRIC REVERSE BI-DERIVATIONS IN SEMIPRIME RINGS I

Let R be a semiprime ring, U a square-closed Lie ideal of R and D : R R ! R a symmetric reverse bi-derivation and d be the trace of D: In the present paper, we shall prove that R commutative ring if any one of the following holds: i) d(U) = (0); ii)d(U) Z; iii)[d (x) ; y] 2 Z; iv)d(x)oy 2 Z; v)d ([x; y])[d(x); y] 2 Z; vi)d (x y)(d(x)y) 2 Z; vii)d ([x; y])d(x)y 2 Z viii)d (x y) [d(x); y] 2 Z; ix)d(x) y [d(y); x] 2 Z; x)d([x; y]) (d(x) y) [d(y); x] 2 Z xi)[d(x); y] [g(y); x] 2 Z; for all x; y 2 U; where G : R R ! R is symmetric reverse bi-derivations such that g is the trace of

A result related to derivations on unital semiprime rings

The purpose of this paper is to prove the following result. Let n≥3 be some fixed integer and let R be a (n+1)!2n-2-torsion free semiprime unital ring. Suppose there exists an additive mapping D: R→ R satisfying the relation for all x ∈ R. In this case D is a derivation. The history of this result goes back to a classical result of Herstein, which states that any Jordan derivation on a 2-torsion free prime ring is a derivation.

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

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.

Symmetric generalized biderivations on prime rings

The purpose of the present paper is to prove some results concerning symmetric generalized biderivations on prime and semiprime rings which partially extend some results of Vukman \cite {V}. Infact we prove that: let $R$ be a prime ring of characteristic not two and $I$ be a nonzro ideal of $R$. If $\Delta$ is a symmetric generalized biderivation on $R$ with associated biderivation $D$ such that $[\Delta(x,x), \Delta(y,y)]=0$ for all $x,y \in I$, then one of the following conditions hold\\ \begin{enumerate} \item $R$ is commutative. \item $\Delta$ acts as a left bimultiplier on $R$. \end{enumerate}

A pair of generalized derivations in prime, semiprime rings and in 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.