= Solution
The <Nakai–Moishezon criterion> says that a <Cartier divisor> $D$ on a <projective scheme> is <ample> exactly when
$$
\boxed{D^{\dim V}\cdot[V]>0\quad\text{for every positive-dimensional integral closed subvariety }V.}
$$
In particular the test includes each positive-dimensional irreducible component. Here and below, “proper” means proper over the ground <field>, not necessarily a strict subset of $X$. Interpreting it as a strict subset would make the later criteria false even for an integral <projective curve>, whose strict closed <subvarieties> have dimension zero.
The <intersection product> of <Cartier divisors> with cycles depends only on their <numerical equivalence of divisors> classes. One way to see this is to intersect all but one factor first, obtaining a one-cycle; replacing the remaining factor by a numerically equivalent divisor does not change its pairing with that cycle. Multilinearity then handles replacement of every factor. Consequently $D\equiv D'$ makes all the displayed numbers equal. Thus
$$
\boxed{D\equiv D'\Longrightarrow(D\text{ ample}\iff D'\text{ ample}).}
$$
The criterion applies to the integral reduced <subvarieties> of a possibly <nonreduced scheme>; <ampleness on reduced components> explains why the nilpotent structure does not change this condition.
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