Generalized derivations on Lie ideals in prime rings

Basudeb Dhara; Sukhendu Kar; Sachhidananda Mondal

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 1, page 179-190
  • ISSN: 0011-4642

Abstract

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Let R be a prime ring with its Utumi ring of quotients U and extended centroid C . Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F ( u ) [ F ( u ) , u ] n = 0 for all u ∈ L , where n ≥ 1 is a fixed integer. Then one of the following holds:

there exists λ ∈ C \lambda \in C such that F ( x ) = λ x F(x)=\lambda x for all x ∈ R x\in R ;

R R satisfies s 4 s_4 and F ( x ) = a x + x b F(x)=ax+xb for all x ∈ R x\in R , with a , b ∈ U a, b\in U and a - b ∈ C a-b\in C ;

char ( R ) = 2 \mathop {\rm char}(R)=2 and R R satisfies s 4 s_4 .

 As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.

How to cite

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Dhara, Basudeb, Kar, Sukhendu, and Mondal, Sachhidananda. "Generalized derivations on Lie ideals in prime rings." Czechoslovak Mathematical Journal 65.1 (2015): 179-190. <http://eudml.org/doc/270040>.

@article{Dhara2015,
abstract = {Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F(u)[F(u),u]^n=0$ for all $u \in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.},
author = {Dhara, Basudeb, Kar, Sukhendu, Mondal, Sachhidananda},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime ring; derivation; generalized derivation; extended centroid; Utumi quotient ring; Lie ideal; Banach algebra; generalized derivations; Lie ideals; prime rings; Banach algebras; extended centroid; Utumi quotient rings},
language = {eng},
number = {1},
pages = {179-190},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized derivations on Lie ideals in prime rings},
url = {http://eudml.org/doc/270040},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Dhara, Basudeb
AU - Kar, Sukhendu
AU - Mondal, Sachhidananda
TI - Generalized derivations on Lie ideals in prime rings
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 179
EP - 190
AB - Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F(u)[F(u),u]^n=0$ for all $u \in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.
LA - eng
KW - prime ring; derivation; generalized derivation; extended centroid; Utumi quotient ring; Lie ideal; Banach algebra; generalized derivations; Lie ideals; prime rings; Banach algebras; extended centroid; Utumi quotient rings
UR - http://eudml.org/doc/270040
ER -

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