Matrices having nonzero outer inverses

It is well known that every nonzero von Neumann regular m× n-matrix A over an arbitrary ring R has a nonzero outer inverse n× 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.