Some Anti-triangular Block Ring Group Inverse Matrix

The research on the generalized inverse of matrices over complex felds andskew felds is abundant, so what is the more general ring that these results on thecorresponding research over is an worth problem. The purpose of this paper is tostudy the group inverse of matrices over right Ore domains and associative rings.Let R be an associative ring with unity1. A ring is called a right Ore domainif it possesses no zero divisors and every two elements of the ring have a rightcommon multiple. Rn×nbe the set of all n×n matrices over R. A square matrixA is said to be group invertible if there exists X such that AXA=A, XAX=Xand AX=XA, we write X=A. It is well known that A exists, it is unique.Firstly, the paper introduced relative the backgrand of group inverse of matricesand its current advance at home and abroad, and outline some relative defnations ofmatrices over rings and right Ore domains, then we present the main work of paper.It includes the representation and existence of the group inverse of two classes of2×2block matrix over rings, and some examples are given; the existence andrepresentation of group inverse of three classes block matrix over right Ore domainsare given, and give examples. The results generalize some current relative works.