  We introduce the decomposition rank, a notion of covering dimension for
nuclear C^*-algebras. The decomposition rank generalizes ordinary covering
dimension and has nice permanence properties; in particular, it behaves well
with respect to direct sums, quotients, inductive limits, unitization and
quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is
invariant under stabilization. It turns out that the decomposition rank can be
finite only for strongly quasidiagonal C^*-algebras and that it is closely
related to the classification program.
