Additive Injective Maps Preserving Inverses Of Matrices On Full Matrix Module Over Local Rings

Linear preserver problem(LPP) is a very active topic in the field of matrix theory, which concerns the maps and operators preserving some invariant of matrices. It has been widely applied in the differential equations, systems control and other fields. In the recent decades, the limits on LPP has been weakened, such as the linear condition has been substituted by additive or some other ones.After introducing the background and the development of preserver problem, we study the problem of additive maps preserving inverses of matrices from matrix modules onto matrix modules over commutative local rings. The main results obtained in this thesis are as follows:Let R be a commutative local ring with identity and2,3∈R*. Then f is an additive injective map from Mn(R) to Mn(R) that preserves inverses of matrices, if and only if f is one of the following two formsi) there exits P∈GLn(R) such that for all A∈Mn(R), f(A)=εPAδP-1, where ε=±1.ⅱ) there exits P∈GLn (R) such that for all A∈Mn (R), f(A)=εP(AT)δP-1, where ε=±1, AT is the transpose of matrix A. In ⅰ) and ⅱ),δ is an injective endomorphism of R.