On a Lie ring of generalized inner derivations

Abstract

In this paper, we define a set including of all $f_{a}$ with $a\in R$ generalized derivations of $R\ $and is denoted by $f_{R}.$ It is proved that (i) the mapping $g:L\left( R\right) \rightarrow f_{R}$ given by $g\left( a\right) =f_{-a}$ for all $a\in R$ is a Lie epimorphism with kernel $N_{\sigma,\tau};$ (ii) if $R$ is a semiprime ring and $\sigma$ is an epimorphism of $R$, the mapping $h:f_{R}\rightarrow I\left( R\right) $ given by $h\left( f_{a}\right) =i_{\sigma\left( -a\right) }$ is a Lie epimorphism with kernel $l\left( f_{R}\right) ;$ (iii) if $f_{R}$ is a prime Lie ring and $A,B$ are Lie ideals of $R,$ then $\left[ f_{A},f_{B}\right] =\left( 0\right) $ implies that either $f_{A}=\left( 0\right) $ or $f_{B}=\left( 0\right)$.