  Noncommutative (NC) sphere is introduced as a quotient of the enveloping
algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we
introduce projective modules which are analogues of line bundles on the usual
sphere (we call them quantum line bundles) and define a multiplicative
structure in their family. Also, we compute a pairing between certain quantum
line bundles and finite dimensional representations of the NC sphere in the
spirit of the NC index theorem. A new approach to constructing the differential
calculus on a NC sphere is suggested. The approach makes use of the projective
modules in question and gives rise to a NC de Rham complex being a deformation
of the classical one.
