  The goal of this paper is to establish some facts concerning the problem of
zeroing the output of an input-output system that does not have relative
degree. The approach taken is to work with systems that have a Chen-Fliess
series representation. The main result is that a class of generating series
called primely nullable series provides the building blocks for solving this
problem using the shuffle algebra. It is shown that the shuffle algebra on the
set of generating polynomials is a unique factorization domain so that any
polynomial can be uniquely factored modulo a permutation into its irreducible
elements for the purpose of identifying nullable factors. This is achieved
using the fact that this shuffle algebra is isomorphic to the symmetric algebra
over the vector space spanned by Lyndon words. A specific algorithm for
factoring generating polynomials into its irreducible factors is presented
based on the Chen-Fox-Lyndon factorization of words.
