Abstract
Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]
| Original language | English |
|---|---|
| Article number | 1850046 |
| Journal | Journal of Algebra and its Applications |
| Volume | 17 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2018 Mar 1 |
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All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Applied Mathematics
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An Engel condition with b -generalized derivations for Lie ideals. / Liau, Pao Kuei; Liu, Cheng-Kai.
In: Journal of Algebra and its Applications, Vol. 17, No. 3, 1850046, 01.03.2018.Research output: Contribution to journal › Article
TY - JOUR
T1 - An Engel condition with b -generalized derivations for Lie ideals
AU - Liau, Pao Kuei
AU - Liu, Cheng-Kai
PY - 2018/3/1
Y1 - 2018/3/1
N2 - Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]
AB - Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]
UR - http://www.scopus.com/inward/record.url?scp=85017010381&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85017010381&partnerID=8YFLogxK
U2 - 10.1142/S0219498818500469
DO - 10.1142/S0219498818500469
M3 - Article
AN - SCOPUS:85017010381
VL - 17
JO - Journal of Algebra and Its Applications
JF - Journal of Algebra and Its Applications
SN - 0219-4988
IS - 3
M1 - 1850046
ER -