A class of algebras called down-up algebras was introduced by G. Benkart and T. Roby (1998, J. Algebra 209, 305-344). We classify the finite dimensional simple modules over Noetherian down-up algebras and show that in some cases every finite dimensional module is semisimple. We also study the question of when two down-up algebras are isomorphic.

A generalization of down-up algebras was introduced by Cassidy and Shelton in [11], the so-called generalized down-up algebras. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X 2 − αX − β are not both roots of unity.

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In this paper we consider a partial action α of a polycyclic by finite group G on a ring R. We prove that if R is right noetherian, then the partial skew group ring R α G is also right noetherian. Extending the methods of Passman in Passman (Trans Am Math Soc 301:737-759, 1987), we obtain a description of the prime spectrum of R α G. The results obtained are applied to get bounds for the Krull dimension and the classical Krull dimension of R α G.

Abstract. We study injective hulls of simple modules over differential operator rings R[θ; d], providing necessary conditions under which these modules are locally Artinian. As a consequence we characterize Ore extensions ofsuch that injective hulls of simple S-modules are locally Artinian.