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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 $f_{B}=\left( 0\right)$.
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참고문헌 (5)
참고문헌 신청
B. Brown / 1958 / Prime ideals in nonassociative rings / Trans. Amer. Math. Soc 89 (1) : 245 ~ 255
C. R. Jordan / 1978 / Lie rings of derivations of associative rings / J. London Math. Soc. (2) 17 (1) : 33 ~ 41
F. Ali / 2011 / On a Lie ring of generalized derivations of non-commutative rings / Int. J. Algebra 5 (8) : 397 ~ 402
M. Bresar / 1991 / On the distance of the composition of two derivations to the generalized derivations / Glasgow Math. J 33 (1) : 89 ~ 93
N. Jacobson / 1937 / Abstract derivation and Lie algebras / Trans. Amer. Math. Soc 42 (2) : 206 ~ 224
C. R. Jordan / 1978 / Lie rings of derivations of associative rings / J. London Math. Soc. (2) 17 (1) : 33 ~ 41
F. Ali / 2011 / On a Lie ring of generalized derivations of non-commutative rings / Int. J. Algebra 5 (8) : 397 ~ 402
M. Bresar / 1991 / On the distance of the composition of two derivations to the generalized derivations / Glasgow Math. J 33 (1) : 89 ~ 93
N. Jacobson / 1937 / Abstract derivation and Lie algebras / Trans. Amer. Math. Soc 42 (2) : 206 ~ 224
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