摘要 : We introduce the notion of integrality of Grothendieck cat-egories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of lo-cally noetherian schemes. Two different spaces assoc... 展开 We introduce the notion of integrality of Grothendieck cat-egories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of lo-cally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integral -ity, 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 cat-egory and deduce Goldie's theorem on the existence of the quotient ring as its consequence. (c) 2021 Elsevier Inc. All rights reserved. 收起