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 |
|---|---|
| Pages (from-to) | 905-915 |
| Number of pages | 11 |
| Journal | Linear and Multilinear Algebra |
| Volume | 59 |
| Issue number | 8 |
| DOIs | |
| Publication status | Published - 2011 Aug 1 |
Fingerprint
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
Cite this
}
Strong commutativity preserving generalized derivations on Lie ideals. / Liu, Cheng Kai; Liau, Pao Kuei.
In: Linear and Multilinear Algebra, Vol. 59, No. 8, 01.08.2011, p. 905-915.Research output: Contribution to journal › Article
TY - JOUR
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.
UR - http://www.scopus.com/inward/record.url?scp=79961150104&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79961150104&partnerID=8YFLogxK
U2 - 10.1080/03081087.2010.535819
DO - 10.1080/03081087.2010.535819
M3 - Article
AN - SCOPUS:79961150104
VL - 59
SP - 905
EP - 915
JO - Linear and Multilinear Algebra
JF - Linear and Multilinear Algebra
SN - 0308-1087
IS - 8
ER -