  The purpose of this paper is to study finiteness conditions on injective
hulls of simple modules over Noetherian Down-Up algebras. We will show that the
Noetherian Down-Up algebras A(\alpha,\beta,\gamma) which are fully bounded are
precisely those which are module-finite over a central subalgebra. We show that
injective hulls of simple A(\alpha,\beta,\gamma)-modules are locally Artinian
provided the roots of X^2-\alpha X-\beta are distinct roots of unity or both
equal to one.
