Matrices having nonzero outer inverses

Abstract

It is well known that every nonzero von Neumann regular $m\times n$-matrix $A$ over an arbitrary ring $R$ has a nonzero outer inverse $n\times m$-matrix $B$ in the sense that $B=BAB$. Generalizing previous work on von Neumann regular matrices, the matrices having nonzero outer inverses over semiperfect rings are characterized as the matrices having some entry outside the Jacobson radical of $R$. Such matrices over finite semiperfect rings and finite commutative rings are counted, and several applications are given.