A quadratic differential identity with skew derivations

Yarbil, Nihan Baydar; Filippis, Vincenzo De

doi:10.1080/00927872.2017.1316853

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“…The study of derivations on prime rings goes back to 1957 by Posner, see [23]. A variety of results have been motivated by this work [2], [5], [6]. A well known theorem of Posner (see [23]) states that if R is a prime ring and d a nonzero derivation of R such that [d(x), x] ∈ Z(R) for all x ∈ R, then R must be commutative.…”

Section: Introductionmentioning

confidence: 99%

Annihilators of skew derivations with Engel conditions on prime rings

Pehlivan

Albaş

2019

Czech.Math.J.

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Section: Introductionmentioning

confidence: 99%

Annihilators of skew derivations with Engel conditions on prime rings

Pehlivan

Albaş

2019

Czech.Math.J.

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Pair of generalized derivations acting on multilinear polynomials in prime rings

Dhara¹,

Kar

Das

2020

Hacettepe Journal of Mathematics and Statistics

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Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C and f (r 1 , . . . , r n ) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that G ̸ = Id (identity map) andThen one of the following holds:(1) there exist λ ∈ C and µ ∈ C such that F (x) = λx and G(x) = µx for all x ∈ R with 2µ = 1;(2) there exist λ ∈ C and p, q ∈ U such that F (x) = λx and G(x) = px + xq for allx ∈ R with p + q ∈ C, 2(p + q) = 1 and f (x 1 , . . . , x n ) 2 is central valued on R;(3) there exist λ ∈ C and a ∈ U such that F (x) = [a, x] and G(x) = λx for all x ∈ R with f (x 1 , . . . , x n ) 2 is central valued on R; (4) there exist λ ∈ C and a, b ∈ U such that F (x) = ax + xb and G(x) = λx for allx ∈ R with a + b ∈ C, 2λ = 1 and f (x 1 , . . . , x n ) 2 is central valued on R; (5) there exist a, p ∈ U such that F (x) = xa and G(x) = px for all x ∈ R, with (p − 1)a = −ap ∈ C and f (x 1 , . . . , x n ) 2 is central valued on R; (6) there exist a, q ∈ U such that F (x) = ax and G(x) = xq for all x ∈ R with a(q − 1) = −qa ∈ C and f (x 1 , . . . , x n ) 2 is central valued on R.

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A quadratic differential identity with skew derivations

Cited by 2 publications

References 19 publications

Annihilators of skew derivations with Engel conditions on prime rings

Annihilators of skew derivations with Engel conditions on prime rings

Pair of generalized derivations acting on multilinear polynomials in prime rings

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