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Let $R$ be a 2-torsion free prime ring and let $σ, τ$ be automorphisms of $R$ . For any $x, y \in R$ , set ${[x, y]}_{σ, τ} = x σ (y) - τ (y) x$ . Suppose that $d$ is a $(σ, τ)$ -derivation defined on $R$ . In the present paper it is shown that $(i)$ if $R$ satisfies ${[d (x), x]}_{σ, τ} = 0$ , then either $d = 0$ or $R$ is commutative $(i i)$ if $I$ is a nonzero ideal of $R$ such that $[d (x), d (y)] = 0$ , for all $x, y \in I$ , and $d$ commutes with both $σ$ and $τ$ , then either $d = 0$ or $R$ is commutative. $(i i i)$ if $I$ is a nonzero ideal of $R$ such that $d (x y) = d (y x)$ , for all $x, y \in I$ , and $d$ commutes with $τ$ , then $R$ is commutative. Finally a related result has been obtain for $(σ, τ)$ -derivation....

On $τ$ -centralizers of semiprime rings.

Albaş, Emine (2007)

Sibirskij Matematicheskij Zhurnal

Order completions of semiprime rings

Walter D. Burgess, Robert M. Raphael (1981)

Czechoslovak Mathematical Journal

Orthogonal generalized $(σ, τ)$ -derivations of semiprime rings.

Gölbaşi, Öznur, Aydin, Neşet (2007)

Sibirskij Matematicheskij Zhurnal

$P (r, m)$ near-rings.

Balakrishnan, R., Suryanarayanan, S. (2000)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

Posner's second theorem and an annihilator condition.

De Filippis, V., Di Vincenzo, O.M. (2001)

Mathematica Pannonica

Posner's second theorem and annihilator conditions with generalized skew derivations

Vincenzo De Filippis, Feng Wei (2012)

Colloquium Mathematicae

Let be a prime ring of characteristic different from 2, $_{r}$ be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds: (a) there exists λ ∈ such that (x) = λx for all x ∈ ; (b) there exist $q \in_{r}$ and...

Potenzen von invarianten Untergruppen in Ringen mit Involution

Walter Streb (1986)

Rendiconti del Seminario Matematico della Università di Padova

Prime and semiprime rings with symmetric skew n-derivations

Ajda Fošner (2014)

Colloquium Mathematicae

Let n ≥ 3 be a positive integer. We study symmetric skew n-derivations of prime and semiprime rings and prove that under some certain conditions a prime ring with a nonzero symmetric skew n-derivation has to be commutative.

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On some equations related to derivations in rings.

On strong commutativity-preserving maps.

On strongly prime semiring.

On the compactness of minimal spectrum

On the generalized hypercentralizer of a Lie ideal in a prime ring

On the Structure of Certain PP-Rings.

On von Neumann regular rings. VII.

On zero subrings and periodic subrings.

On $(σ, τ)$ -derivations in prime rings