Linear Maps Preserving The Relation Of Equality Of Rank On Symmetric Matrix Module

In this thesis, we study the linear maps of preserving the relation of equality ofrank on the module of symmetric matrices over principal ideal domain (PID). Themain results obtained in this thesis are as follows:(1) we characterize the linear maps of preserving rank-1on the module of sym-metric matrices over PID.(2) we characterize the linear surjective maps of preserving preserving the relationof equality of rank on the module of symmetric matrices over PID.