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On bilinear maps on matrices with applications to commutativity preservers
Brešar, Matej ; Šemrl, Peter
Let ▫$M_n$▫ be the algebra of all ▫$n \times n$▫ matrices over a commutative unital ring ▫$\mathcal{C}$▫, and let ▫$\mathcal{L}$▫ be a ▫$\mathcal{C}$▫-module. Various characterizations of bilinear ...maps ▫$\{\,.\,,\,.\,\}: M_n \times M_n \to \mathcal{L}$▫ with the property that ▫$\{x,y\} = 0$▫ whenever ▫$x$▫ any ▫$y$▫ commute are given. As the main application of this result we obtain the definitive solution of the problem of describing (not necessarily bijective) commutativity preserving linear maps from ▫$M_n$▫ into ▫$M_n$▫ for the case where ▫$\mathcal{C}$▫ is an arbitrary field; moreover, this description is valid in every finite dimensional central simple algebra.
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