Zero product determined matrix algebras
Zero product determined matrix algebras
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DOI：
10.1016/j.laa.2007.11.018
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发表时间：
2009-03
影响因子：
1.1
通讯作者：
M. Brešar;M. Grasic;J. Ortega
中科院分区：
文献类型：
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作者：
M. Brešar;M. Grasic;J. Ortega
设A是交换单位环C上的代数。我们说A是零积确定的，如果对每个C-模X和每个双线性映射{·，·}：A×A→X，下列成立：如果{x，y}=0，当xy=0时，则存在线性算子T使得对所有x，y∈A，{x，y}=T（xy）.如果我们在这个定义中用Lie（resp. Jordan）乘积，则我们说A是零Lie（resp. Jordan）产品确定。证明了矩阵代数Mn（B），n 2，其中B是任意的单位代数，总是零积确定的，并且在一定的技术限制下也是零Jordan积确定的.本文主要讨论Mn（B）是否为零Lie积的问题。我们证明了这并不对所有的单位代数B都成立。然而，如果B是确定的零Lie积，则Mn（B）也是如此。
Let A be an algebra over a commutative unital ring C. We say that A is zero product determined if for every C-module X and every bilinear map {·,·}:A×A→X the following holds: if {x,y}=0 whenever xy=0, then there exists a linear operator T such that {x,y}=T(xy) for all x,y∈A. If we replace in this definition the ordinary product by the Lie (resp. Jordan) product, then we say that A is zero Lie (resp. Jordan) product determined. We show that the matrix algebra Mn(B), n⩾2, where B is any unital algebra, is always zero product determined, and under some technical restrictions it is also zero Jordan product determined. The bulk of the paper is devoted to the problem whether Mn(B) is zero Lie product determined. We show that this does not hold true for all unital algebras B. However, if B is zero Lie product determined, then so is Mn(B).