An equivalence relation is defined and studied on the set of $B$-stable ideals in the nilradical of the Lie algebra of a Borel subgroup $B$. Techniques are developed to compute the equivalence relation and these are carried out in the exceptional groups. There is a natural partial order on equivalence classes coming from inclusion of one ideal in another. A main theorem is that this partial order is a refinement of the closure ordering on nilpotent orbits.