  Tropical ideals, introduced in arXiv:1609.03838, define subschemes of
tropical toric varieties. We prove that the top-dimensional parts of their
varieties are balanced polyhedral complexes of the same dimension as the ideal.
This means that every subscheme of a tropical toric variety defined by a
tropical ideal has an associated class in the Chow ring of the toric variety. A
key tool in the proof is that specialization of variables in a tropical ideal
yields another tropical ideal; this plays the role of hyperplane sections in
the theory. We also show that elimination theory (projection of varieties)
works for tropical ideals as in the classical case. The matroid condition that
defines tropical ideals is crucial for these results.
