Strongly regular matrices revisited

Abstract

We prove a necessary condition and a sufficient condition for an n×n-matrix A with determinantal rank ρ(A)=t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its tth compound matrix Ct(A). In particular, a non-zero n×n-matrix A with ρ(A)=t over a local commutative ring R is strongly regular if and only if Tr(Ct(A)) is a unit in R, and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras.