Generalized inverses over integral domains. II. Group inverses and Drazin inverses

Abstract

This is a continuation of an earlier paper by the authors on generalized inverses over integral domains. The main results consist of necessary and sufficient conditions for the existence of a group inverse, a new formula for a group inverse when it exists, and necessary and sufficient conditions for the existence of a Drazin inverse. We show that a square matrix A of rank r over an integral domain R has a group inverse if and only if the sum of all r × r principal minors of A is an invertible element of R. We also show that the group inverse of A when it exists is a polynomial in A with coefficients from R.