Inverse-preserving Operators On Upper Triangular Matrix Algebra

In this thesis, we introduce the background and the development of Preserver Problems firstly. Then we study the bijective map of preserving inverse of upper trian-gular matrix algebra over non-commutative local rings and division rings respectively. The main results obtained are as follows:1. Let R be a non-commutative local ring with2,3invertible. we characterize the inverse-preservering linear bijection∫of upper triangular matrix spaces over R, and obtain that∫has the following form: f(A)=εQAσQ-1,(?)A∈Tn(R), where ε=±1, Q is product of finite numbers of pemutation matrices and one diagonal matrix over R, a is an automorphism of R.2. Let D be a division ring, we give a complete description for inverse-preservering additive bijection∫of upper triangular matrix algebra over D, and obtain that∫has the following form: f(A)=eQA(TQ-1,(?)A∈Tn(D), where ε=±1, Q is product of finite numbers of pemutation matrices and one diagonal matrix over D, σ is an automorphism of D.