Zero product determined matrix algebras

被引:49
作者
Bresar, Matej [1 ]
Grasic, Mateja [1 ]
Sanchez Ortega, Juana [2 ]
机构
[1] Univ Maribor, FNM, Dept Math & Comp Sci, SLO-2000 Maribor, Slovenia
[2] Univ Malaga, Dept Algebra Geometria & Topol, E-29071 Malaga, Spain
关键词
Zero product determined algebra; Zero Lie product determined algebra; Zero Jordan product determined algebra; Matrix algebra; Bilinear map; Linear map; MAPS;
D O I
10.1016/j.laa.2007.11.018
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 x 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 is an element of 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 M-n(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 M-n(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 M-n(B). (c) 2007 Elsevier Inc. All rights reserved.
引用
收藏
页码:1486 / 1498
页数:13
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