  We introduce a generalization of the notion of a negligible morphism and
study the associated tensor ideals and thick ideals. These ideals are defined
by considering deformations of a given monoidal category $\mathcal{C}$ over a
local ring $R$. If the maximal ideal of $R$ is generated by a single element,
we show that any thick ideal of $\mathcal{C}$ admits an explicitely given
modified trace function. As examples we consider various Deligne categories and
the categories of tilting modules for a quantum group at a root of unity and
for a semisimple, simply connected algebraic group in prime characteristic. We
prove an elementary geometric description of the thick ideals in quantum type A
and propose a similar one in the modular case.
