Ideal structure of multiplier algebras of simple C*-algebras with real rank zero

We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of the multiplier algebra M(A), is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of M(A) modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of M(A) is reflected in the fact that M(A) can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.