  We introduce the notion of integrality of Grothendieck categories as a
simultaneous generalization of the primeness of noncommutative noetherian rings
and the integrality of locally noetherian schemes. Two different spaces
associated to a Grothendieck category yield respective definitions of
integrality, and we prove the equivalence of these definitions using a
Grothendieck-categorical version of Gabriel's correspondence, which originally
related indecomposable injective modules and prime two-sided ideals for
noetherian rings. The generalization of prime two-sided ideals is also used to
classify locally closed localizing subcategories. As an application of the main
results, we develop a theory of singular objects in a Grothendieck category and
deduce Goldie's theorem on the existence of the quotient ring as its
consequence.
