Commuting Maps On Matrix Rings Over Integral Domains

Matrix theory is one of the most important branches in algebra. It has veryextensive applications in many different ways, such as, economics, control theory,graph theory, and so on. Studying matrix preserving problem has become a veryactive research area in matrix theory.In this thesis, we first introduce the background and the present developmentof matrix preserving problems, then we study the commuting maps on matrix ringsover integral domains and obtain the following four results:(1) We obtain the commuting maps on singular matrices over integral domainsand study the application to a preserving problem about derivation. At the sametime, we give the forms of commuting maps on other matrices over integraldomains.(2) We characterize the commuting maps on rank-n matrices over integraldomains of order more than2.(3) We obtain the commuting maps on rank-k matrices over integral domainsof characteristic not2or3. At the same time, the commuting maps on rank-1matrices over integral domains have the same structure.(4) We characterize the commuting maps on rank-2matrices over integraldomains.