| Preserver problem is a very active topic in the field of matrix theory, which concerns the maps and operators that preserve certain relations, subsets and properties. It has wide applications in the differential equations, systems control and other fields, so that the study on preserver problems have achieved fruitful results.After introducing the background and the development of preserver problems, we study the problem of additive maps preserving inverses of matrices from symmetric matrix modules Sn(R) to full matrix modules Mn(R) over commutative local rings. The main result obtained in this thesis is as follows:Let R be a commutative local ring with identity,2,3∈R*. Then f is an additive injective map from Sn(R) to Mn(R) that preserves inverses of matrices, if and only if there exits P∈GLn(R) such that for all A∈Sn(R), f(A)=±AAδP-1Where δ is an injective endomorphism of R. |
