  We propose notions of "Noetherian" and "integral" for schemes over an abelian
symmetric monoidal category $(\mathcal C,\otimes,1)$. For Noetherian integral
schemes, we construct a "function field" that is a commutative monoid object of
$(\mathcal C,\otimes,1)$. Under certain conditions, we show that a Noetherian
scheme over $(\mathcal C,\otimes,1)$ is integral if and only if it is reduced
and irreducible.
