Generalized inverses of matrices over commutative rings (Q1338509)

Let \({\mathcal A}\) be a commutative ring with identity and \(A\) an \(m \times n\) matrix over \(A\). \(A\) is called regular if there is a matrix \(G\) such that \(AGA = A\). A matrix \(A\) of rank \(r\) is Rao-regular if there exist \(\{c^ J_ I\}\) in \({\mathcal A}\) such that \[ \left( \sum_{IJ} c^ J_ I | A^ I_ J | \right) A = A \] where \(I\) and \(J\) are \(r\)- element subsets of \(\{1,2, \dots, m\}\) and \(\{1,2, \dots, n\}\), respectively, and \(A^ J_ I\) is a submatrix of \(A\) determined by rows indexed by \(I\) and columns indexed by \(J\). The quantity \(\sum_{IJ} c^ J_ I | A^ I_ J |\) is called a Rao-idempotent of \(A\). It is proved that \(A\) is regular if and only if it is a sum of Rao- regular matrices with mutually orthogonal Rao idempotents. Furthermore, necessary and sufficient conditions are given for a matrix to have a group inverse over a commutative ring.