Linear Maps Preserving Tripotence From Symmetric Matrix Modules Onto Matrix Modules Over Local Rings

Preserving problems concern the characterization of maps between matrix spaces that preserve some invariants. Studying preserving problems can help us to understand matrix invariants, functions, sets and relations. According to the property of maps, preserving problems can be divided into three categories, that is, linear preserving problems, additive preserving problems and general preserving problems. According to the property of invariants, preserving problems can be divided into four categories, that is, preserving subsets, preserving relations, preserving functions and preserving transformations. Linear preserver problem is an active topic in the field of matrix theory, it characterizes the linear operators which preserve certain functions, subsets, relations or transformations invariants between matrix sets. Linear preserver problem has wide applications in other areas, such as differential equations, systems control, etc. In the recent years, the study on the linear preserver problem has made great progress.After introducing the background and development of the linear preserver problem, we study the problem of linear maps preserving tripotence from symmetric matrix modules onto matrix modules over local rings. The main results obtained in this thesis are as follows:Let R be a commutative local ring, let and be positive integers with n≤m.Supposen f is a linear map from n×n symmetric matrix modules to matrix algebra S_n(R)m×M_m(R) over R which preserves tripotence. In this thesis, we describe the linear maps preserving tripotence of matrices from S_n( R) to M_m(R).