  It is proved that every separable $C^*$-algebra of real rank zero contains an
AF-sub-$C^*$-algebra such that the inclusion mapping induces an isomorphism of
the ideal lattices of the two $C^*$-algebras and such that every projection in
a matrix algebra over the large $C^*$-algebra is equivalent to a projection in
a matrix algebra over the AF-sub-$C^*$-algebra. This result is proved at the
level of monoids, using that the monoid of Murray-von Neumann equivalence
classes of projections in a $C^*$-algebra of real rank zero has the refinement
property. As an application of our result, we show that given a unital
$C^*$-algebra $A$ of real rank zero and a natural number $n$, then there is a
unital $^*$-homomorphism $M_{n_1} \oplus ... \oplus M_{n_r} \to A$ for some
natural numbers $r,n_1, ...,n_r$ with $n_j \ge n$ for all $j$ if and only if
$A$ has no representation of dimension less than $n$.
