In Addition To The Ring On The Matrix Space Preserving Rank 1 The Export Mapping And Applications,

In recent forty years, the matrix preserver problems as an important role of al-gebra has an wide applications in many areas, because it has a great theoretical andpractical signifcance, and many of the problems in diferential equations, systemcontrol, the felds of mathematical statistical has a broad background in practicalapplications. In preserver problems, rank-1preservers are the core of many pre-serving invariants in the study feld, many scholars have done a great job in thisarea. So, we have gained lot of results on rank-1preserver problems in recent years.The considerations of the present paper were inspired by it, and studied producedmap of preserving rank-1(preserving rank k) in general matrices space and Hermitiematrices on division ring. Let D be a division ring, and f_ijbe maps from D toitself. Denote by V a set of matrices over D. If a map f: Vâ†'V is defned byAâ†'(f_ij(aij)) for any A=(aij)∈V, then we say that f is induced by {f_ij}. WhenV=Mmn(D), H_n(D)(i.e., sets of m×n matrices and Hermitian matrices of ordern over D if D has an involution), this paper determines the general forms of all mapsf induced by {f_ij} preserving rank-1matrices. As applications, the general form ofall maps f induced by {f_ij} preserving rank k is given.