Additive Injective Maps Preserving Inverses Of Hermitian Matrices

Preserver problem is a very important and fertile topic in the field of matrix theory, and a great deal of results in this area have been given for the past several decades.For the matrix over a commutative local ring equipped an involution Hermitian matrix has a good structure. So we study the problem of additive maps preserving inverses of matrices from Hermitian matrix modules Hn(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 equipped an involution and2,3∈R*Then f is an additive injective map from Hn(R) to Mn(R) that preserves inverses of matrices, if and only if there exits P∈GLn (R) such that for all A∈Hn (R), f(A)=±PAδP-1Where δ is an injective endomorphism of R.