Reduced scheme

A scheme is reduced when every local ring ${\mathcal{O}}_{X,x}$ is a reduced ring, equivalently when its structure sheaf has no nonzero nilpotent element. The affine scheme $\mathrm{Spec}A$ is reduced exactly when $A$ is reduced.

An integral scheme is a nonempty scheme that is both reduced and irreducible. An affine scheme $\mathrm{Spec}A$ is integral exactly when $A$ is an integral domain.