Integral Non-commutative Spaces

A non-commutative space X is a Grothendieck category Mod X. We say X is integral if there is an indecomposable injective X-module X such that its endomorphism ring is a division ring and every X-module is a subquotient of a direct sum of copies of X. A noetherian scheme is integral in this sense if and only if it is integral in the usual sense. We show that several classes of non-commutative spaces are integral. We also define the function field and generic point of an integral space and show that these notions behave as one might expect.