# Strong commutativity preserving generalized derivations on Lie ideals

Cheng Kai Liu, Pao Kuei Liau

Research output: Contribution to journalArticle

13 Citations (Scopus)

### Abstract

We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f:L → R is a map and g is a generalized derivation of R such that [f(x), g(y)]=[x, y] for all x, y ∈ L. Then there exist a nonzero α ∈ C and a map μ: L → C such that g(x)=α x for all x ∈ R and f (x)=α-1x+ μ(x) for all x ∈ L, except when R ⊆ M2(F), the 2 × 2 matrix ring over a field F.

Original language English 905-915 11 Linear and Multilinear Algebra 59 8 https://doi.org/10.1080/03081087.2010.535819 Published - 2011 Aug 1

### Fingerprint

Lie Ideal
Generalized Derivation
Commutativity
Differential Identity
Elementary matrix
Extended Centroid
Matrix Computation
Matrix Ring
Prime Ring

### All Science Journal Classification (ASJC) codes

• Algebra and Number Theory

### Cite this

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title = "Strong commutativity preserving generalized derivations on Lie ideals",
abstract = "We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f:L → R is a map and g is a generalized derivation of R such that [f(x), g(y)]=[x, y] for all x, y ∈ L. Then there exist a nonzero α ∈ C and a map μ: L → C such that g(x)=α x for all x ∈ R and f (x)=α-1x+ μ(x) for all x ∈ L, except when R ⊆ M2(F), the 2 × 2 matrix ring over a field F.",
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In: Linear and Multilinear Algebra, Vol. 59, No. 8, 01.08.2011, p. 905-915.

Research output: Contribution to journalArticle

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T1 - Strong commutativity preserving generalized derivations on Lie ideals

AU - Liu, Cheng Kai

AU - Liau, Pao Kuei

PY - 2011/8/1

Y1 - 2011/8/1

N2 - We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f:L → R is a map and g is a generalized derivation of R such that [f(x), g(y)]=[x, y] for all x, y ∈ L. Then there exist a nonzero α ∈ C and a map μ: L → C such that g(x)=α x for all x ∈ R and f (x)=α-1x+ μ(x) for all x ∈ L, except when R ⊆ M2(F), the 2 × 2 matrix ring over a field F.

AB - We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f:L → R is a map and g is a generalized derivation of R such that [f(x), g(y)]=[x, y] for all x, y ∈ L. Then there exist a nonzero α ∈ C and a map μ: L → C such that g(x)=α x for all x ∈ R and f (x)=α-1x+ μ(x) for all x ∈ L, except when R ⊆ M2(F), the 2 × 2 matrix ring over a field F.

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