  This paper concerns the computation and identification of the (homological)
Conley index over the integers, in the context of discrete dynamical systems
generated by continuous maps. We discuss the significance with respect to
nonlinear dynamics of using integer, as opposed to field, coefficients. We
translate the problem into the language of commutative ring theory. More
precisely, we relate shift equivalence in the category of finitely generated
abelian groups to the classification of $\mathbb{Z}[t]$-modules whose
underlying abelian group is given. We provide tools to handle the
classification problem, but also highlight the associated computational
challenges.
