  We describe a general framework for prime, completely prime, semiprime, and
primitive ideals of an abelian 2-category. This provides a noncommutative
version of Balmer's prime spectrum of a tensor triangulated category. These
notions are based on containment conditions in terms of thick subcategories of
an abelian category and thick ideals of an abelian 2-category. We prove
categorical analogs of the main properties of noncommutative prime spectra.
Similar notions, starting with Serre subcategories of an abelian category and
Serre ideals of an abelian 2-category, are developed. They are linked to Serre
prime spectra of $\mathbb{Z}_+$-rings. As an application, we construct a
categorification of the quantized coordinate rings of open Richardson varieties
for symmetric Kac-Moody groups, by constructing Serre completely prime ideals
of monoidal categories of modules of the KLR algebras, and by taking Serre
quotients with respect to them.
