Generalized skew derivations characterized by acting on zero products
Generalized skew derivations characterized by acting on zero products
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DOI：
10.2140/pjm.2004.216.293
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发表时间：
2004-10
影响因子：
0.6
通讯作者：
Tsiu-Kwen Lee
中科院分区：
文献类型：
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作者：
Tsiu-Kwen Lee
设A是素环，它的对称Martindale商环包含一个非平凡的幂等元。然后，通过作用于零积刻划了A的广义斜导子。准确地说，如果g；-：a！A是可加映射，使得对所有x，ae(X)g(Y)+-(X)y=0；y2A，其中ae是A的自同构，则g和-都刻画为A的非零理想上的特殊广义ae-导子。B称为导子，如果对所有x；y2 A都有-(Xy)=-(X)y+x-(Y)。在最近的一篇论文中，Lu和Li证明了如下结果(6，定理6)：设B是Banach空间X中包含单位算子i的标准算子代数，并且-：B！B是一个线性映射，使得对任意对A，-(AB)=-(A)B+A-(B)；对于AB=0的B2B。则-(AB)=-(A)B+A-(B)I A-(I)B对所有A；B 2B。此外，如果加法-(I)=0，则-是导数。结果表明，如果标准算子代数上的加法映射满足零积对元上导子的展开式，则它几乎是导子。由于标准算子代数包含许多幂等元，从这个角度出发，KE和P.-H.Lee在素环的背景下研究了作用在零积上的映射(2)。为了给出它的准确陈述，我们首先修正一些记号。自始至终，除非特别说明，否则A总是表示具有中心Z、扩张质心C和对称Martindale商环Q的素环。
Let A be a prime ring such that its symmetric Martindale quotient ring contains a nontrivial idempotent. Then generalized skew derivations of A are characterized by acting on zero products. Precisely, if g;-:A ! A are additive maps such that ae(x)g(y) + -(x)y = 0 for all x;y 2 A with xy = 0 where ae is an automorphism of A, then both g and - are characterized as specific generalized ae-derivations on a nonzero ideal of A. 1. Results Let B be a ring with a subring A. An additive map -:A ! B is called a derivation if -(xy) = -(x)y + x-(y) for all x;y 2 A. In a recent paper Jing, Lu and Li proved the following result (6, Theorem 6): Let B be a standard operator algebra in a Banach space X containing the identity operator I, and -:B ! B be a linear map such that -(AB) = -(A)B + A-(B) for any pair A;B 2 B with AB = 0. Then -(AB) = -(A)B + A-(B) i A-(I)B for all A;B 2 B. Moreover, if in addition -(I) = 0, then - is a derivation. The result says that an additive map on a standard operator algebra is almost a derivation if it satisfies the expansion formula of derivations on pair elements with zero product. Since standard operator algebras involve many idempotents, from this point of view Chebotar, Ke and P.-H. Lee studied maps acting on zero products in the context of prime rings (2). To give its precise statement we first fix some notation. Throughout, unless specially stated, A always denotes a prime ring with center Z, extended centroid C and symmetric Martindale quotient ring Q. Moreover, let