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Additive surjections preserving rank one and applications. (English) Zbl 1075.15007

The authors characterize epimorphisms of the additive group of the full matrix algebra \(M_n(F)\) over a field \(F\), preserving rank one. Every such epimorphism \(f\) is uniquely determined by two automorphisms of the vector space \(F^n\) and a field automorphism \(g\), i.e. there exist matrices \(P,Q\in\text{GL}_n(F)\) such that either \(f(A)= PA^gQ\) or \(f(A)= P(A^g)^T Q\), where \(A^g= (g(a_{ij}))\) for \(A= (a_{ij})\).

MSC:

15A04 Linear transformations, semilinear transformations
15A03 Vector spaces, linear dependence, rank, lineability
15A30 Algebraic systems of matrices
16S50 Endomorphism rings; matrix rings
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