On Jordan ideals in prime rings with generalized derivations
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
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 $R$.
Abstract
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 $R$.
참고문헌 (7)
- R. Awtar, "Lie and jordan structure in prime rings with derivations", Proc. Amer. Math. Soc, 41, pp.67~74 (1973)
- M. Breˇsar, "On the distance of the composition of two derivations to the generalized derivations", Glasgow Math. J, 33(1), pp.89~93 (1991)
- M. Breˇsar, "Centralizing mappings and derivations in prime rings", J. Algebra, 156(2), pp.385~394 (1993)
- M. El-Soufi, "Generalized derivations on Jordan ideals in prime rings", Turkish J. Math, 38(2), pp.233~239 (2014)
- L. Oukhtite, "Generalized derivations centralizing on Jordan ideals of rings with involution", Turkish J. Math, 38(2), pp.225~232 (2014)
- E. C. Posner, "Derivations in prime rings", Proc. Amer. Math. Soc, 8, pp.1093~1100 (1957)
- S. M. A. Zaidi, "On Jordan ideals and left (θ; θ)-derivations in prime rings", Int. J. Math. Math. Sci, 2004(37-40), pp.1957~1964 (2004)