Integral scheme

Remarks

A scheme is integral if and only if it is reduced and its underlying space is irreducible. An affine scheme $\operatorname{Spec} A$ is integral if and only if $A$ is an integral domain. Then its generic point is $(0)$, and the local ring there is the field of fractions of $A$, the function field. Example: $\operatorname{Spec} \Z$ is integral, since $\Z$ is a domain and each $\mathcal{O}(D(n)) = \Z[1/n]$ is a domain. Non-example: $\operatorname{Spec} k[x, y]/(xy)$ for a field $k$ is reduced but not integral, since $x y = 0$ with $x \ne 0$ and $y \ne 0$. The open set $U = \emptyset$ is excluded because $\mathcal{O}_X(\emptyset)$ is the zero ring, which is not a domain.