Zero modules for systems over rings

This study centers on some of the results given by F. Bostwick et al. (J. Control Optim., vol.25, no.1, p.86-99, Jan. 1987) which connect zeros of the Rosenbrock system matrix with zeros of the transfer function. It is shown that some of these results can be obtained for linear time-invariant systems over an arbitrary ring. In particular, the Ω-zero and Γ-zero modules for systems over arbitrary rings are defined. The kernel and cokernel of the transfer function, not necessarily proper, and the Rosenbrock system map are shown to be canonically isomorphic. These isomorphisms are embedded into a commutative exact diagram. The corresponding results extended to the Ω-level and to the Γ-level are then examined