The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra H is studied. Such a Hopf algebra H is an extension of an affine commutative Hopf algebra A by a finite dimensional Hopf algebra $\overline{H}$ . The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $\overline{H}^{\ast}$ by the finite dual $A^{\circ}$ of A. This decomposition is then further analysed using the Cartier–Gabriel–Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$ . The detailed consequences for a number of families of examples are then studied.

Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.

Assume R is an associative ring with unity. Let $(K, \leq _p)$ be an AEC such that $K \subseteq R\text {-Mod}$

and K is closed under finite direct sums, then:

• • If K is closed under pure-injective envelopes, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $ .

• • If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $ .

• • Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $ .

As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.

Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of $\mathfrak {s}$ -torsion modules.

This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.

A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer ${{C}_{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $S$ of the Schrödinger (Lie) algebra. The simple ${{C}_{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $S$ (over the centre). It is proved that some (prime) factor algebras of $S$ and ${{C}_{S}}(H)$ are tensor homological$/$Krull minimal.

We show that every finitely generated algebra that is a finitely generated module over a finitely generated commutative subalgebra is an automaton algebra in the sense of Ufnarovskii.

A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.

We construct an interesting family of connected graded domains of Gel’fand–Kirillov dimension 4, and show that the general member of this family is noetherian. The algebras we construct are Koszul and have global dimension 4. They fail to be Artin–Schelter Gorenstein, however, showing that a theorem of Zhang and Stephenson for dimension 3 algebras does not extend to dimension 4. The Auslander–Buchsbaum formula also fails to hold for these algebras. The algebras we construct are birational to ℙ2, and their existence disproves a conjecture of the first author and Stafford. The algebras can be obtained as global sections of a certain quasicoherent graded sheaf on ℙ1×ℙ1, and our key technique is to work with this sheaf. In contrast to all previously known examples of birationally commutative graded domains, the graded pieces of the sheaf fail to be ample in the sense of Van den Bergh. Our results thus require significantly new techniques.

Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains that are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl–Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups ${\mathcal O}_q(\text{GL}_n)$ and ${\mathcal O}_q(\text{SL}_n)$.

A complete structural characterization of submonoids $S$ of a polycyclic-by-finite group such that the semigroup algebra $K[S]$ over a field $K$ is right noetherian is obtained. It follows that such algebras are also left noetherian.

Let $R$ be a semi-prime Noetherian ring of injective dimension 1. Let $P$ be a minimal prime ideal of $R$. In this paper it is shown that $R/P$ need not have injective dimension 1. Necessary and sufficient conditions are given for $R/P$ to have injective dimension 1.

The question is addressed of when all pure-projective modules are direct sums of finitely presented modules. It is proved that this is the case over hereditary noetherian rings. Partial results are obtained for uniserial rings. Some of the methods are model-theoretic, and the techniques developed using these may be of interest in their own right.

Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. These results are unified and extended to the much wider class of rings with Auslander dualizing complexes.

The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

AS a generalization of Morita duality, Kraemer introduced the notion of quasi-duality and showed that each left linearly compact ring has a quasi-duality. Let R be an associative ring with identity and R[[x]] the power series ring. We prove that (1) R[[x]] has a quasi-duality if and only if R has a quasi-duality; (2) R[[x]] is left linearly compact if and only if R is left linearly compact and left noetherian; and (3) R[[x]] has a Morita duality if and only if R is left noetherian and has a Morita duality induced by a bimodule RUS such that S is right noetherian.

Let R be a ring and M a right R-module. Let σ[M] be the full subcategory of Mod-R subgenerated by M. An M-natural class 𝒦 is a subclass of σ[M] closed under submodules, direct sums, isomorphic copies, and M-injective hulls. We present some equivalent conditions each of which describes when σ has the property that direct sums of (M-)injective modules in σ are (M-)injective. Specializing to particular M, and/or special subclasses we obtain many new results and known results as corollaries.