We obtained a “decomposition scheme” of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsidó), 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,∞, the largest type II finite ideal A
II,1, the largest type II essentially infinite ideal A
II,∞, and the largest type III ideal A
III of any C*-algebra A such that A
d,1 + A
d,∞ + A
II,1 + A
II,∞ + 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 ideal spaces, 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.