Additive preservers of rank on alternate matrix spaces over fields and applications
Let us denote by \(F\) any field and by \(A^t\) the transpose of any matrix \(A\) with entries in \(F\). A square matrix \(A\) is said to be \(alternate\) if \(A^t=-A\) and all diagonal elements are zeros. These matrices play a considerable role in the theory of quadratic forms and classical groups, and therefore the endomorphisms preserving their properties take on great interest. The author determines the general form of all additive preservers of rank, rank-additivity and rank-subtractivity. Assume that \(K_n (F )\) is the set of all \(n \times n,n\geq 4\) alternate matrices over \(F\) ; then \((K_n (F) , + , \cdot ) \) is a non-associative ring under the usual addition ``\(+\)'' and the multiplication ``\(\cdot\)'' defined in the following way: \(X \cdot Y=XYX \) for all \( X , Y \in K_n(F)\). The author characterizes all ring endomorphisms of \((K_n (F),+ , \cdot)\), i.e. all the additive maps \(\phi\) such that \(\phi (X \cdot Y ) = \phi (X) \cdot \phi (Y)\) for any \(X , Y \in K_n (F)\).
scientific article; zbMATH DE number 2165365