Injective Modules over Down-Up Algebras

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(α,β,γ) which are fully bounded are precisely those which are module-finite over a central subalgebra. We show that injective hulls of simple A(α,β,γ)-modules are locally Artinian provided the roots of X^2-αX-βare distinct roots of unity or both equal to one.

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