  We express the multigraded Betti numbers of an arbitrary monomial ideal in
terms of the multigraded Betti numbers of two basic classes of ideals. This
decompo- sition has multiple applications. In some concrete cases, we use it to
construct minimal resolutions of classes of monomial ideals; in other cases, we
use it to compute projective dimensions. To illustrate the effectiveness of the
structural decomposition, we give a new proof of a classic theorem by
Charalambous.
