  The Corona Factorization Property of a C*-algebra, originally defined to
study extensions of C*-algebras, has turned out to say something important
about intrinsic structural properties of the C*-algebra. We show in this paper
that a \sigma-unital C*-algebra A of real rank zero has the Corona
Factorization roperty if and only if its monoid V(A) of Murray-von Neumann
equivalence classes of projections in matrix algebras over A has a certain
(rather weak) comparability property that we call the Corona Factorization
Property (for monoids). We show that a projection in such a C*-algebra is
properly infinite if (and only if) a multiple of it is properly infinite.
  The latter result is obtained from some more general result we establish
about conical refinement monoids. We show that the set of order units (together
with the zero-element) in a conical refinement monoid is again a refinement
monoid under the assumption that the monoid satisfies weak divisibility; and if
u is an element in a refinement monoid such that nu is properly infinite, then
u can be written as a sum u = s + t such that ns and nt are properly infinite.
