  We study a noncommutative gauge theory on a fuzzy four-sphere. The idea is to
use a matrix model with a fifth-rank Chern-Simons term and to expand matrices
around the fuzzy four-sphere which corresponds to a classical solution of this
model. We need extra degrees of freedom since algebra of coordinates does not
close on the fuzzy four-sphere. In such a construction, a fuzzy two sphere is
added at each point on the fuzzy four-sphere as extra degrees of freedom. It is
interesting that fields on the fuzzy four-sphere have higher spins due to the
extra degrees of freedom. We also consider a theory around the north pole and
take a flat space limit. A noncommutative gauge theory on four-dimensional
plane, which has Heisenberg type noncommutativity, is considered.
