环上某些2×2块阵的Drazin逆

The Drazin inverses of some 2 × 2 block matrices over rings

令R是有1的结合环,Rm×n是R上所有m×n矩阵的集合,若正整数k及A∈Rn×n,满足方程组Ak+1X=Ak,XAX=X,AX=XA,则称X为A的Drazin逆,当k=1时,A#=AD被称为A的群逆。在一般环上研究此问题,给出环上三类2×2块阵的Drazin逆的存在性条件及表示。

Let R be an associative ring with 1, R^m×n be the sets of all the m×n matrices over R, for the positive integer k and the matrix A in R^n×n , matrix X is called the Drazin inverse of A if X satisfies the equations, A^k＋1X = A^k, XAX = X and AX = XA. Denote A^# = A^D when k = 1 , and A^# is called the group inverse of A. The existence conditions and the representation of Drazin inverse for three class of 2 × 2 block matrices over rings are given.