On Automorphisms and Commutativity in Semiprime Rings

Liu, Cheng-Kai

doi:10.4153/cmb-2011-185-5

Cited by 12 publications

(3 citation statements)

References 32 publications

(24 reference statements)

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“…Several other related generalizations can been founded in [15][16][17][18][19][20]. The goal of this paper is to extend Theorem L to b-generalized derivations.…”

Section: Introduction and Resultsmentioning

confidence: 98%

An Engel condition withb-generalized derivations

Liu

2016

Linear and Multilinear Algebra

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Let R be a noncommutative prime ring with the extended centroid C, I a nonzero ideal of R and g a b-generalized derivation of R. We show that, if [g(x m ), x n ] k = 0 for all x ∈ I, where m, n, k are fixed positive integers, then there exists λ ∈ C such that g(x) = λx for all x ∈ R unless R ∼ = M 2 (GF(2)), the 2 × 2 matrix ring over the Galois field GF(2) of two elements. This gives a natural generalization of the results for derivations, generalized derivations and generalized σ -derivations with an X-inner automorphism σ .

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“…Several other related generalizations can been founded in [15][16][17][18][19][20]. The goal of this paper is to extend Theorem L to b-generalized derivations.…”

Section: Introduction and Resultsmentioning

confidence: 98%

An Engel condition withb-generalized derivations

Liu

2016

Linear and Multilinear Algebra

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“…], x n k ] = 0 for all x in a one-sided ideal of prime rings. Since then many related Engel type identities with derivations or automorphisms have been investigated in the literature (see [2,22,26,29] and the references therein). Precisely, Lanski [20] and Lee [24] proved the following result.…”

Section: Introduction and Resultsmentioning

confidence: 99%

An Engel condition with skew derivations for one-sided ideals

Liu

2015

Monatsh Math

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“…Recently, some authors have obtained commutativity of prime and semiprime rings with derivations, automorphisms, generalized derivations et al satisfying certain polynomial constraints (cf. ; [2,6,12,14,16,17] and references therein). In [10], Daif and Bell showed that if in a semiprime ring R there exists a nonzero ideal I of R and a derivation d such that d([x, y]) ± [x, y] = 0 for all x, y ∈ I, then I ⊆ Z(R).…”

Section: Introductionmentioning

confidence: 99%

On generalized derivations and commutativity of prime gamma-rings

Huang¹,

Daif²

2016

GJOM

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Let M be a prime Γ-ring with center Z(M), I a nonzero ideal of M and F be a generalized derivation with associated nonzero derivation d. In the present paper, our purpose is to produce commutativity results for prime Γ-rings M admitting a generalized derivation F satisfying any one of the properties: (i) F(xαy)∓xαy ∈ Z(M), (ii) F(xαy)∓yαx ∈ Z(M), (iii) F(x)αF(y)∓xαy ∈ Z(M), (iv) F(x)αF(y)∓yαx ∈ Z(M), (v) F([x, y]α)=[F(x), y]α, (vi) [F(x), y]α∓[x, F(y)]α=0, (vii) F([x, y]α)∓[d(x), d(y)]α=0, for all x,y ∈ I and α ∈ Γ. Also, some examples are given to show that the primeness of the various results is not superfluous.

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On Automorphisms and Commutativity in Semiprime Rings

Abstract: Abstract. Let R be a semiprime ring with center Z(R). For x, y ∈ R, we denote by [x, y] = xy − yx the commutator of x and y.

Cited by 12 publications

References 32 publications

An Engel condition withb-generalized derivations

An Engel condition withb-generalized derivations

An Engel condition with skew derivations for one-sided ideals

On generalized derivations and commutativity of prime gamma-rings

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