  Let $R$ be a commutative ring. A quasi-Gorenstein $R$-module is an $R$-module
such that the grade of the module and the projective dimension of the module
are equal and the canonical module of the module is isomorphic to the module
itself. After discussing properties of finitely generated quasi-Gorenstein
modules, it is shown that this definition allows for a characterization of
diagonal matrices of maximal rank over a Noetherian factorial domain $R$
extending a theorem of Frobenius and Stickelberger to modules of projective
dimension 1 over a commutative Noetherian factorial domain.
