  Let $\psi: A \to A'$ be a cyclic contraction of dimer algebras, with $A$
non-cancellative and $A'$ cancellative. $A'$ is then prime, noetherian, and a
finitely generated module over its center. In contrast, $A$ is often not prime,
nonnoetherian, and an infinitely generated module over its center. We present
certain Morita equivalences that relate the representation theory of $A$ with
that of $A'$.
  We then characterize the Azumaya locus of $A$ in terms of the Azumaya locus
of $A'$, and give an explicit classification of the simple $A$-modules
parameterized by the Azumaya locus. Furthermore, we show that if the smooth and
Azumaya loci of $A'$ coincide, then the smooth and Azumaya loci of $A$
coincide. This provides the first known class of algebras that are
nonnoetherian and infinitely generated modules over their centers, with the
property that their smooth and Azumaya loci coincide.
