The aim of this paper is to introduce a notion of zero dynamics for linear, time-delay systems. To this aim, we use the correspondence between time-delay systems and system with coefficients in a ring, so to exploit algebraic and geometric methods. By combining the algebraic notion of zero module and the geometric structure of the lattice of invariant submodules of the state module, we point out a natural way to define zero dynamics in particular situations. Then, we extend our approach, in order to encompass more general situations, and we provide a definition of zero dynamics which applies to a number of interesting cases, including that of time-delay systems with commensurable delays. Relations between this notion and fixed poles in closed loop control schemes, as well as with a concept of phase minimality, are discussed.