The cover polynomial C(D) = C(D;x, y) of a digraph D is a twovariable polynomial whose coefficients are determined by the number of vertex coverings of D by directed paths and cycles. Just as for the Tutte polynomial for undirected graphs (cf. [11, 16]), various properties of D can be read off from the values of C(D;x, y). For example, for an n-vertex digraph D, C(D; 1, 0) is the number of Hamiltonian paths in D, C(D; 0, 1) is the permanent of adjacency matrix of D, and C(D; 0,−1) is (−1)n times the determinant of the adjacency matrix of D. In this paper, we extend these ideas to a much more general setting, namely, to matrices with elements taken from an arbitrary commutative ring with identity. In particular, we establish a reciprocity theorem for this generalization, as well as establishing a symmetric function version of the new polynomial, similar in spirit to Stanley’s symmetric function generalization [13] of the chromatic polynomial of a graph, and Tim Chow’s symmetric function generalization [5] of the usual cover polynomial. We also show that all of the generalized polynomials and symmetric functions can also be obtained by a deletion/contraction process.