  We study how geometric properties of tropical convex sets and polytopes,
which are of interest in many application areas, manifest themselves in their
algebraic structure as modules over the tropical semiring. Our main results
establish a close connection between pure dimension of tropical convex sets,
and projectivity (in the sense of ring theory). These results lead to a
geometric understanding of idempotency for tropical matrices. As well as their
direct interest, our results suggest that there is substantial scope to apply
ideas and techniques from abstract algebra (in particular, ring theory) in
tropical geometry.
