Bezout Whole District Group Inverse Matrix

The research on the generalized inverses of matrices over commutative rings, especially the condition of regularity of matrix(For a matrix A, if there exists matrix X, such that AX A=A, then we say A is regularity), is abundant; see references therein. However, the research on the generalized inverses of matrices over non-commutative rings is relatively sparse. The purpose of this paper is to study the group inverse of matrices over an important associative ring-Bezout domain, and generalize some results which are well known and given at present.If every finitely generated left (right) ideal in a non-zero ring R which has a unit element 1 and no zero divisors is principal, then R is called a Bezout domain. Integral Ring, Polynomial ring in an indeterminate over field, Division Ring, non-commutative P.I.D., and Valuation Ring and so on are Bezout domain. For a matrix A∈Rn×n, We say that A# is the group inverse of A if A# is a common solution of the matrix equations: AXA= A,XAX=X,AX=XA. It is easy to prove that if A# exists then A# is unique.This paper is divided into four sections. In the first section, we outline the relative definitions of the group inverse of matrix and the Bezout domain, and introduce the study of the group inverse of matrix over rings. In the second section, we study the existence of the group inverse for matrix A, obtaining several equivalent conditions, as well as the existence and representation of the group inverse for 2 x 2 block upper triangular matrix. This generalizes the relative results. In the third section, we study the group inverses of products of two matrices, obtain the condition for the reverse order law of group inverses, and generalize the relative results. In the fourth section, we study the existence and representation of the group inverse for two classes 2×2 block matrices, generalizing the relative results. All results we obtain are new even for commutative principal ideal domains.