ИСТИНА

The first results on transformations preserving matrix invariants is due to Frobenius. This result describes the structure of linear maps T preserving the determinant function, i.e., detX = detT(X) for all X. Later on there were several extension of this result which are due to Diedonnie, Schur, Dynkin and others. Later on different maps preserving matrix properties, invariants, relations on operator or matrix algebras over various algebraic structures were investigated by many researchers. We plan to discuss the corresponding problems for matrices over division rings and characterize semilinear maps preserving Dieudonn ́e determinant and singularity. This provides non-commutative analogs of Frobenius and Dieudonn ́e theorems.