## Taiwanese Journal of Mathematics

### GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

#### Abstract

Let $R$ be a noncommutative primering with its Utumi ring of quotients $U$, $C=Z(U)$ the extendedcentroid of $R$, $F$ a generalized derivation of $R$ and $I$ anonzero ideal of $R$. Suppose that there exists $0\neq a\in R$ such that $a(F([x,y])^n-[x,y])=0$ for all $x,y \in I$, where $n\geq 1$ is a fixedinteger. Then either $n=1$ and $F(x)=bx$ for all $x\in R$ with$a(b-1)=0$ or $n\geq 2$ and one of the following holds:

1. char $(R)\neq 2$, $R\subseteq M_2(C)$, $F(x)=bx$ for all$x\in R$ with $a(b-1)=0$ (In this case $n$ is an odd integer);

2. char $(R)= 2$, $R\subseteq M_2(C)$ and $F(x)=bx+[c,x]$ forall $x\in R$ with $a(b^n-1)=0$.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 3 (2015), 943-952.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133671

Digital Object Identifier
doi:10.11650/tjm.19.2015.4043

Mathematical Reviews number (MathSciNet)
MR3353262

Zentralblatt MATH identifier
1357.16057

#### Citation

Dhara, Basudeb; De Filippis, Vincenzo; Pradhan, Krishna Gopal. GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS. Taiwanese J. Math. 19 (2015), no. 3, 943--952. doi:10.11650/tjm.19.2015.4043. https://projecteuclid.org/euclid.twjm/1499133671

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