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Let $R$ be a $2$-torsion free prime ring and $J$ be a nonzero Jordan ideal of $R$. Let $F$ and $G$ be two generalized derivations with associated derivations $f$ and $g$, respectively. Our main result in this paper shows that if $F(x)x-xG(x)=0$ for all $x\in J$, then $R$ is commutative and $F=G$ or $ G$ is a left multiplier and $F=G+f$. This result with its consequences generalize some recent results due to El-Soufi and Aboubakr in which they assumed that the Jordan ideal $J$ is also a subring of $R$.
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참고문헌 (7)
참고문헌 신청
R. Awtar / 1973 / Lie and jordan structure in prime rings with derivations / Proc. Amer. Math. Soc 41:67~74
M. Breˇsar / 1991 / On the distance of the composition of two derivations to the generalized derivations / Glasgow Math. J 33(1):89~93
M. Breˇsar / 1993 / Centralizing mappings and derivations in prime rings / J. Algebra 156(2):385~394
M. El-Soufi / 2014 / Generalized derivations on Jordan ideals in prime rings / Turkish J. Math 38(2):233~239
L. Oukhtite / 2014 / Generalized derivations centralizing on Jordan ideals of rings with involution / Turkish J. Math 38(2):225~232
M. Breˇsar / 1991 / On the distance of the composition of two derivations to the generalized derivations / Glasgow Math. J 33(1):89~93
M. Breˇsar / 1993 / Centralizing mappings and derivations in prime rings / J. Algebra 156(2):385~394
M. El-Soufi / 2014 / Generalized derivations on Jordan ideals in prime rings / Turkish J. Math 38(2):233~239
L. Oukhtite / 2014 / Generalized derivations centralizing on Jordan ideals of rings with involution / Turkish J. Math 38(2):225~232
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