设 A 是一个结合环,G ∈ A. G称 为 A 的一个全可导点,如果每一个在G点可导的可加映射φ:A→A (即对任意的S,T∈A 有φ(ST) =φ(S)T +Sφ(T) 且ST=G) 都是一个导子.本文证明了一类三角环上的每个非零元都是全可导点.作为此结果的推论得到:一类域上的三角矩阵环的每个非零元都是全可导点.
Abstract
Let A be an associative ring. Let G ∈ A. We say that G is an allderivable point of A if every derivable additive map φ : A → A at G (i.e., φ(ST) = φ(S)T +Sφ(T) for all S, T ∈ A with ST = G) is a derivation. The aim of the paper is to show that every nonzero element in certain triangular rings is an all-derivable point. As a corollary we prove that every nonzero element in upper triangular matrix rings over a certain field is an all-derivable point.
关键词
全可导点 /
三角环 /
套代数
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Key words
all-derivable point /
triangular rings /
nest algebras
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参考文献
[1] An R. L., Hou J. C., Characterizations of derivations on triangular rings: Additive maps derivable at idempotents, Linear Algebra Appl., 2009, 431(2): 1070–1080.
[2] Cheung W. S., Commuting maps of triangular algebras, J. London Math. Soc., 2001, 63(1): 117–127.
[3] Cheung W. S., Lie derivations of triangular algebras, Linear Multilinear Algebra, 2003, 51(1): 299–310.
[4] Hou J. C., Qi X. F., Characterizations of derivations of Banach space nest algebras: all-derivable points, Linear Algebra Appl., 2010, 432(8): 3183–3200.
[5] Wang Y., Additivity of multiplicative maps on triangular rings, Linear Algebra Appl., 2011, 434(1): 625–635.
[6] Zhu J., Characterization of all-derivable points in nest algebras, Proc. Amer. Math. Soc., 2013, 141(4): 2343–2350.
[7] Zhu J., Xiong C. P., Zhang R. Y., All-derivable points in the algebra of all upper triangular matrices, Linear Algebra Appl., 2008, 429(4): 804–818.
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脚注
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基金
上海市自然科学基金资助项目(14ZR1431200)
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