  The integral group rings $\mathbb{Z}G$ for finite groups $G$ are precisely
those fusion rings whose basis elements have Frobenius-Perron dimension 1, and
each is categorifiable in the sense that it arises as the Grothendieck ring of
a fusion category. Here we analyze the structure and representation theory of
fusion rings with a basis of elements whose Frobenius-Perron dimensions take
exactly one value distinct from 1. Our goal is a set of results which assist in
characterizing when such fusion rings are categorifiable. As proof of concept,
we completely classify the categorifiable near-group fusion rings for an
infinite collection of finite abelian groups, a task that to-date has only been
completed for three such groups.
