  Let $Z$ be the center of a nonnoetherian dimer algebra $A$ on a torus. We
show that the nilradical $\operatorname{nil}Z$ of $Z$ is prime, may be nonzero,
and consists precisely of the central elements that vanish under a cyclic
contraction of $A$. This implies that the nonnoetherian scheme
$\operatorname{Spec}Z$ is irreducible. We also show that the reduced center
$\hat{Z} = Z/\operatorname{nil}Z$ embeds into the center $R$ of the
corresponding ghor algebra, and that their normalizations are equal. Finally,
we give three characterizations of the normality of $R$, and show that if
$\hat{Z}$ is normal, then it has the special form $k + J$ where $J$ is an ideal
of the cycle algebra of $A$.
