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We give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of 
$\sigma $-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 
$\mathcal{M}(A)$, is therefore analyzed. In important cases it is shown that, if $A$ has finite scale then the quotient of $\mathcal{M}(A)$ modulo any closed ideal 
$I$ that properly contains $A$ has stable rank one. The intricacy of the ideal structure of $\mathcal{M}(A)$ is reflected in the fact that $\mathcal{M}(A)$ can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.
