  Underlying a general noncommutative algebra with both noncommutative
coordinates and noncommutative momenta in a (1+1)-dimensional spacetime, a
chiral boson Lagrangian with manifest Lorentz covariance is proposed by
linearly imposing a generalized self-duality condition on a noncommutative
generalization of massless real scalar fields. A significant property uncovered
for noncommutative chiral bosons is that the left- and right-moving chiral
scalars cannot be distinguished from each other, which originates from the
noncommutativity of coordinates and momenta. An interesting result is that
Dirac's method can be consistently applied to the constrained system whose
Lagrangian explicitly contains space and time. The self-duality of the
noncommutative chiral boson action does not exist.
