  We obtained a "decomposition scheme" of C*-algebras. We show that the classes
of discrete C*-algebras (as defined by Peligard and Zsido), type II C*-algebras
and type III C*-algebras (both defined by Cuntz and Pedersen) form a good
framework to "classify" C*-algebras. In particular, we found that these classes
are closed under strong Morita equivalence, hereditary C*-subalgebras as well
as taking "essential extension" and "normal quotient". Furthermore, there exist
the largest discrete finite ideal $A_{d,1}$, the largest discrete essentially
infinite ideal $A_{d,\infty}$, the largest type II finite ideal $A_{II,1}$, the
largest type II essentially infinite ideal $A_{II,\infty}$, and the largest
type III ideal $A_{III}$ of any C*-algebra $A$ such that $A_{d,1} +
A_{d,\infty} + A_{II,1} + A_{II,\infty} + A_{III}$ is an essential ideal of
$A$. This "decomposition" extends the corresponding one for $W^*$-algebras.
  We also give a closer look at C*-algebras with Hausdorff primitive spectrum,
AW*-algebras as well as local multiplier algebras of C*-algebras. We find that
these algebras can be decomposed into continuous fields of prime C*-algebras
over a locally compact Hausdorff space, with each fiber being non-zero and of
one of the five types mentioned above.
