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 Note on Differential Identities in Prime and Semiprime Rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

Generalized derivations preserving Engel condition in prime and semiprime rings

Let [Formula: see text] be a prime ring with [Formula: see text], [Formula: see text] a non-central Lie ideal of [Formula: see text], [Formula: see text] its Martindale quotient ring and [Formula: see text] its extended centroid. Let [Formula: see text] and [Formula: see text] be nonzero generalized derivations on [Formula: see text] such that [Formula: see text] Then there exists [Formula: see text] such that [Formula: see text] and [Formula: see text], for any [Formula: see text], unless [Formula: see text], where [Formula: see text] is the algebraic closure of [Formula: see text].

On Commutativity of Prime and Semiprime - Rings with Reverse Derivations

Let M be a weak Nobusawa -ring and γ be a non-zero element of Γ. In this paper, we introduce concept of k-reverse derivation, Jordan k-reverse derivation, generalized k-reverse derivation, and Jordan generalized k-reverse derivation of Γ-ring, and γ-homomorphism, anti-γ-homomorphism of M. Also, we give some commutattivity conditions on γ-prime Γ-ring and γ-semiprime Γ-ring .

On Skew Left n-Derivations with Lie Ideal Structure

In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.

On Skew Left n-Derivations with Lie Ideal Structure

In this paper the centralizing and commuting concerning skew left -derivations and skew left -derivations associated with antiautomorphism on prime and semiprime rings were studied and the commutativity of Lie ideal under certain conditions were proved.

On Certain Identities with Automorphisms on Lie Ideals in Prime and Semiprime Rings

Let R be a prime ring of characteristic different from 2, Z(R) its center, L a Lie ideal of R, and m, n, s, t ≥ 1 fixed integers with t ≤ m + n + s. Suppose that α is a non-trivial automorphism of R and let Φ(x, y) = [x, y]t – [x, y]m [α([x, y]),[x, y]]n [x, y]s. Thus, (a) if Φ(u, v) = 0 for any u, v ∈ L, then L ⊆ Z(R); (b) if Φ(u, v) ∈ Z(R) for any u, v ∈ L, then either L ⊆ Z(R) or R satisfies s4, the standard identity of degree 4. We also extend the results to semiprime rings.