Simplicity of iterated Ore extensions

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

Ore Extension Rings Satisfy the Weak PS-Rings

The main result of this paper is that: If R is a weak right PS-ring, then A = R[x; α, δ], the Ore extension ring, is a weak right PS-ring whenever the following conditions hold on R is an (α, δ)-compatible NI-ring with nil(R) nilpotent, α(e) = e and δ(e) = 0 for every idempotent e ∈ R.

Classification of Simple Modules of the Ore Extension $$K[X][Y; f\frac{d}{dX}]$$

For the algebras $$\Lambda $$Λ in the title of the paper, a classification of simple modules is given, an explicit description of the prime and completely prime spectra is obtained, the global and the Krull dimensions of $$\Lambda $$Λ are computed.

Bott–Samelson Varieties and Poisson Ore Extensions

We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

NAKAYAMA AUTOMORPHISMS OF ORE EXTENSIONS OVER POLYNOMIAL ALGEBRAS

Nakayama automorphisms play an important role in the fields of noncommutative algebraic geometry and noncommutative invariant theory. However, their computations are not easy in general. We compute the Nakayama automorphism ν of an Ore extension R[x; σ, δ] over a polynomial algebra R in n variables for an arbitrary n. The formula of ν is obtained explicitly. When σ is not the identity map, the invariant EG is also investigated in terms of Zhang’s twist, where G is a cyclic group sharing the same order with σ.

Some types of ring elements in Ore extensions over noncommutative rings

Let [Formula: see text] be an associative unital ring with an endomorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text]. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the [Formula: see text]-regular elements and also the von Neumann local elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible. As an application, we completely characterize the clean elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible.

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

On Ore extension and skew power series rings with some restrictions on zero-divisors

The study of rings with right Property ([Formula: see text]), has done an important role in noncommutative ring theory. Following literature, a ring [Formula: see text] has right Property ([Formula: see text]) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ([Formula: see text]) of Ore extensions as well as skew power series rings. We will show that if [Formula: see text] is a right duo ring, then the skew power series ring [Formula: see text] has right Property ([Formula: see text]), when [Formula: see text] is right Noetherian and [Formula: see text]-compatible. Moreover, for a right duo ring [Formula: see text] which is [Formula: see text]-compatible, it is shown that (i) the Ore extension ring [Formula: see text] has right Property ([Formula: see text]) and (ii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip. As a corollary of our results, we provide answers to some open questions related to Property [Formula: see text], raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ([Formula: see text]) and their extensions, J. Algebra 315 (2007) 612–628].

On constant Armendariz rings

Let [Formula: see text] be a ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. In this article, we study [Formula: see text]-constant Armendariz rings and radicals of the Ore extension [Formula: see text], in terms of a [Formula: see text]-constant Armendariz ring [Formula: see text] with an [Formula: see text]-condition, are determined. We prove that several properties transfer between [Formula: see text] and the Ore extension [Formula: see text], in case [Formula: see text] is [Formula: see text]-compatible [Formula: see text]-constant Armendariz.