On Jordan and Jordan higher derivable maps of rings
2
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
Let $\mathcal{R}$ be a 2-torsion free unital ring containing a non-trivial idempotent. An additive map $\delta$ from $\mathcal{R}$ into itself is called a Jordan derivable map at commutative zero point if $\delta(AB+BA)=\delta(A)B+B\delta(A)+A\delta(B)+\delta(B)A$ for all $A, B\in\mathcal{R}$ with $AB=BA=0$. In this paper, we prove that, under some mild conditions, each Jordan derivable map at commutative zero point has the form $\delta(A)=\psi(A)+CA$ for all $A\in\mathcal{R}$, where $\psi$ is an additive Jordan derivation of $\mathcal{R}$ and $C$ is a central element of $\mathcal{R}$. Then we generalize the result to the case of Jordan higher derivable maps at commutative zero point. These results are also applied to some operator algebras.
Abstract
Let $\mathcal{R}$ be a 2-torsion free unital ring containing a non-trivial idempotent. An additive map $\delta$ from $\mathcal{R}$ into itself is called a Jordan derivable map at commutative zero point if $\delta(AB+BA)=\delta(A)B+B\delta(A)+A\delta(B)+\delta(B)A$ for all $A, B\in\mathcal{R}$ with $AB=BA=0$. In this paper, we prove that, under some mild conditions, each Jordan derivable map at commutative zero point has the form $\delta(A)=\psi(A)+CA$ for all $A\in\mathcal{R}$, where $\psi$ is an additive Jordan derivation of $\mathcal{R}$ and $C$ is a central element of $\mathcal{R}$. Then we generalize the result to the case of Jordan higher derivable maps at commutative zero point. These results are also applied to some operator algebras.
참고문헌 (16)
- Dominik Benkovič, "Jordan derivations and antiderivations on triangular matrices", Linear Algebra and its Applications, 397, pp.235~244 (2005), Elsevier BV[DOI]
- M. Bresar, "Jordan Derivations on Semiprime Rings", Proceedings of the American Mathematical Society, 104(4), pp.1003~1006 (1988), JSTOR[DOI]
- M. Bresar, "Characterizing homomorphisms, derivations and multipliers in rings with idem-potents", Proc. Roy. Soc. Edinburgh, Sect. A, 137, pp.9~21 (2007)
- Miguel Ferrero, "Higher Derivations and a Theorem By Herstein", Quaestiones Mathematicae, 25(2), pp.249~257 (2002), National Inquiry Services Center (NISC)[DOI]
- Miguel Ferrero, "HIGHER DERIVATIONS OF SEMIPRIME RINGS", Communications in Algebra, 30(5), pp.2321~2333 (2011), Informa UK Limited[DOI]
- I. N. Herstein, "Jordan Derivations of Prime Rings", Proceedings of the American Mathematical Society, 8(6), pp.1104~1110 (1957), JSTOR[DOI]
- C. R. Miers, "Lie homomorphisms of operator algebras", Pacific J. Math, 38, pp.717~735 (1971)
- Madjid Mirzavaziri, "Characterization of Higher Derivations on Algebras", Communications in Algebra, 38(3), pp.981~987 (2010), Informa UK Limited[DOI]
- P. Ribenboim, "Higher derivations of rings. I", Rev. Roumaine Math. Pures Appl, 16, pp.77~110 (1971)
- P. Ribenboim, "Higher derivations of rings. II", Rev. Roumaine Math. Pures Appl, 16, pp.245~272 (1971)
- A. M. Sinclair, "Jordan Homomorphisms and Derivations on Semisimple Banach Algebras", Proceedings of the American Mathematical Society, 24(1), pp.209~214 (1970), JSTOR[DOI]
- Feng Wei, "Generalized Jordan triple higher derivations on semiprime rings", 대한수학회보, 46(3), pp.553~565 (2009), 대한수학회
- Feng Wei, "Higher derivations of triangular algebras and its generalizations", Linear Algebra and its Applications, 435(5), pp.1034~1054 (2011), Elsevier BV[DOI]
- Zhankui Xiao, "Jordan higher derivations on triangular algebras", Linear Algebra and its Applications, 432(10), pp.2615~2622 (2010), Elsevier BV[DOI]
- Jian-Hua Zhang, "Jordan derivations of triangular algebras", Linear Algebra and its Applications, 419(1), pp.251~255 (2006), Elsevier BV[DOI]
- Sha Zhao, "Jordan all-derivable points in the algebra of all upper triangular matrices", Linear Algebra and its Applications, 433(11-12), pp.1922~1938 (2010), Elsevier BV[DOI]