Research On Group Inverse And MP-inverse Over Formal Matrix Ring

As an important expansion of matrix ring,formal matrix ring has been paid attention to by scholars at home and abroad since its proposal.As we all know,the invertible elements in the ring have a very important meaning to the algebraic theory of the ring,and the generalized inverse provides a new way for us to further study the ring theory and the category of modules.Therefore,it is very necessary to study generalized inverses on formal matrix rings.Inspired by this idea,this paper adopts pure algebraic research methods and uses group invertible elements and MP-invertible elements to expand the ring theory of formal matrix rings.This master’s thesis is mainly divided into three chapters.The first chapter mainly gives the background knowledge,basic concepts and symbols about the formal matrix ring Mn(R;s),as well as some basic definitions and related conclusions that the article needs to use.The second chapter mainly studies several properties of the formal matrix ring group inverse and MP-inverse on the associative ring R with identity element 1.Firstly,by characterizing the invertible elements,we show that the matrix A is a group invertible element if and only if σ(A)+I-AA-is invertible,and if and only if σ(A)has a strong clean decomposition.Matrix A is MP-invertible if and only if σ(AA*)+I-AA-is invertible,and if and only if σ(AA*)is strong*-clean decomposition.Secondly,we use the annihilator to further answer the relationship between the formal matrix ring and the direct sum decomposition.A is a group reversible element if and only if Mn(R;s)=0A(?)Mn(R;s)σ(A),and A is an MP-reversible element if and only if Mn(R;s)=°A(?)Mn(R;s)σ(AA*).Finally,we give and prove the necessary and sufficient conditions for the law of absorption and reverse order.The third chapter gives concrete expressions for the group inverse and MPinverse of any matrix in the formal matrix ring Mn(R;s).The main work is to use Schur’s complement theory to trigonometrically decompose the matrix M in the formal matrix ring M2(R;s)under certain conditions,so that M=PAQ,and then obtain the group inverse and MP-inverse expressions.Finally,based on the second chapter,it is obtained and proved that the n-th order matrix in the formal matrix ring Mn(R;s)on the field can find the specific expressions of the group inverse and MP-inverse.