The Drazin inverse of matrices over rings (Q2735491)

Let \(A\) be a square matrix over a commutative ring and let \(k\) be the index of \(A\), so that \(A^k\), \(A^{k+1}\) have the same rank. Suppose there exist matrices \(B,P,Q,P',Q'\) such that \(A^k=PBQ\), \(P'PB=B=BQQ'\). Assume that \(B\) has a group inverse \(B^\#\) and let \(X=BB^\# QAPB+ I-BB^\#\), where \(I\) is the unit matrix. The authors prove that \(A\) has a Drazin inverse \(A^D\) if, and only if, \(X\) is invertible, \(A^D\) being \(PBX^{-1}Q\) when this condition holds. A somewhat similar result is proved in the case where the coefficient ring is non-commutative, conditions like above being imposed on every power of \(A\).