  Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed
under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$
is called a $K$-set if the ideal generated by the coefficients of the elements
of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those
homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a
$K$-set. We also consider corresponding notions of rank of matrices and Betti
numbers of chain complexes; this includes an analysis of the case of McCoy
rank. Our setup also recovers results on jump loci obtained by Kohno and
Pajitnov as a special case.
