Paula A. A. B. Carvalho

Automorphisms of Generalized Down-Up Algebras

A generalization of down-up algebras was introduced by Cassidy and Shelton (2004), 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 − α X − β are not both roots of unity.

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(\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.

MONOLITHIC MODULES OVER NOETHERIAN RINGS

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantized Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained in [arXiv:0906.2930] for down-up algebras.

Injective Hulls of Simple Modules over Differential Operator Rings

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 of S = K[x][θ; σ, d] for σ a K-linear automorphism and d a K-linear σ-derivation of K[x] such that injective hulls of simple S-modules are locally Artinian.

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6] introduced an algebra extension A P [y 1, y 2; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y 1; σ1, δ1][y 2; σ2, δ2]. We also give partial answers to some questions posed in Zhang and Zhang [6].

ON THE AZUMAYA LOCUS OF SOME CROSSED PRODUCTS

Abstract Given D a commutative Noetherian domain and G a finite group acting faithfully on D as automorphisms of D, we describe the Azumaya locus of the crossed product D*G and relate it with the singular locus of DG. To achieve this we will impose some basic homological conditions on D and on D*G. Also we discuss necessary and sufficient conditions for a crossed product R*G, for some ring R, to be Azumaya. Concepts as H-separability and Galois extensions are used.

Simple modules and their essential extensions for skew polynomial rings

Let R be a commutative Noetherian ring and

Ring theoretical properties of affine cellular algebras

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