Skew Polynomial Extensions Research Articles

Left Nil Zero Semicommutative rings

This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices over R is LNZS. Furthermore, it is shown that the polynomial ring over an LNZS may not be LNZS and so is the case of the skew polynomial over an LNZS ring. Therefore, this paper investigates the LNZS property over the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.

Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer

Let k be a field and let A be a finitely generated k-algebra. The algebra A is said to be cancellative if whenever B is another k-algebra with the property that $$A[x]\cong B[x]$$ then we necessarily have $$A\cong B$$ . An important result of Abhyankar, Eakin, and Heinzer shows that if A is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand–Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand–Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar–Eakin–Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.

GENERALISED QUANTUM DETERMINANTAL RINGS ARE MAXIMAL ORDERS

AbstractGeneralised 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.

Hopf Ore Extensions

Brown, O'Hagan, Zhang, and Zhuang gave a set of conditions on an automorphism $\sigma$ and a $\sigma$-derivation $\delta$ of a Hopf $k$-algebra $R$ for when the skew polynomial extension $T=R[x, \sigma, \delta]$ of $R$ admits a Hopf algebra structure that is compatible with that of $R$. In fact, they gave a complete characterization of which $\sigma$ and $\delta$ can occur under the hypothesis that $\Delta(x)=a\otimes x +x\otimes b +v(x\otimes x) +w$, with $a, b\in R$ and $v, w\in R\otimes_k R$, where $\Delta: R\to R\otimes_k R$ is the comultiplication map. In this paper, we show that after a change of variables one can in fact assume that $\Delta(x)=\beta^{-1}\otimes x +x\otimes 1 +w$, with $\beta $ is a grouplike element in $R$ and $w\in R\otimes_k R,$ when $R\otimes_k R$ is a domain and $R$ is noetherian. In particular, this completely characterizes skew polynomial extensions of a Hopf algebra that admit a Hopf structure extending that of the ring of coefficients under these hypotheses. We show that the hypotheses hold for domains $R$ that are noetherian cocommutative Hopf algebras of finite Gelfand-Kirillov dimension.

Effective Buchberger-Zacharias-Weispfenning theory of skew polynomial extensions of subbilateral coherent rings

This work's primary objective is to investigate an effective Buchberger-Zacharias-Weispfenning theory of bimodules over skew polynomial extensions of a coefficient ring satisfying the appropriate notion of coherence. The original contribution of our approach establishes the subbilateral coherence together with the effective computation of subbilateral Gröbner bases for a sufficiently large class of skew polynomial extensions, over term-ordered monoids, of subbilateral Noetherian coherent coefficient rings. The strategy to produce finite bilateral Gröbner bases from subbilateral ones generalizes Kandri-Rody and Weispfenning completion from left Gröbner bases to bilateral ones over a subclass of solvable polynomial rings, as well as Weispfenning completion from Weispfenning restricted modules to bilateral ones over the bivariate Weispfenning extensions of skew fields. It equally results that the class of rings that are both left Noetherian and left coherent (or resp., right Noetherian and right coherent) is closed under iterative skew polynomial extension with bijective conjugation maps.

Nakayama automorphisms of double Ore extensions of Koszul regular algebras

Let $A$ be a Koszul Artin-Schelter regular algebra and $\sigma$ an algebra homomorphism from $A$ to $M_{2\times 2}(A)$. We compute the Nakayama automorphisms of a trimmed double Ore extension $A_P[y_1, y_2; \sigma]$ (introduced in \cite{ZZ08}). Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension $A[t; \theta]$, where $\theta$ is a graded algebra automorphism of $A$. These lead to a characterization of the Calabi-Yau property of $A_P[y_1, y_2; \sigma]$, the skew Laurent extension $A[t^{\pm 1}; \theta]$ and $A[y_1^{\pm 1}, y_2^{\pm 1}; \sigma]$ with $\sigma$ a diagonal type.

From quantum Ore extensions to quantum tori via noncommutative UFDs

All iterated skew polynomial extensions arising from quantized universal enveloping algebras of Kac–Moody algebras are special examples of a very large, axiomatically defined class of algebras, called CGL extensions. For the purposes of constructing initial clusters for quantum cluster algebra structures on an algebra R, and classification of the automorphisms of R, one needs embeddings of R into quantum tori T which have the property that R contains the corresponding quantum affine space algebra A. We explicitly construct such an embedding A⊆R⊂T for each CGL extension R using the methods of noncommutative noetherian unique factorization domains and running a Gelfand–Tsetlin type procedure with normal, instead of central elements. Along the way we classify the homogeneous prime elements of all CGL extensions and we prove that each CGL extension R has an associated maximal torus which covers the automorphisms of R corresponding to all normal elements. For symmetric CGL extensions, we describe the relationship between our quantum affine space algebra A and Cauchon's quantum affine space algebra generated by elements obtained via deleting derivations.

PBW deformations of Artin–Schelter regular algebras

We consider properties and extensions of PBW deformations of Artin–Schelter regular algebras. PBW deformations in global dimension two are classified and the geometry associated to the homogenizations of these algebras is exploited to prove that all simple modules are one-dimensional in the non-PI case. It is shown that this property is maintained under tensor products and certain skew polynomial extensions.

On the Jacobson Radical of Skew Polynomial Extensions of Rings Satisfying a Polynomial Identity

Let R be a ring satisfying a polynomial identity, and let D be a derivation of R. We consider the Jacobson radical of the skew polynomial ring R[x; D] with coefficients in R and with respect to D, and show that J(R[x; D]) ∩ R is a nil D-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when R is commutative.

Connected Hopf algebras and iterated Ore extensions

We investigate when a skew polynomial extension T=R[x;σ,δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand–Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.

Skew Polynomial Extensions over Zip Rings

In this article, we study the relationship between left (right) zip property of and skew polynomial extension over , using the skew versions of Armendariz rings.

QUANTUM UNIQUE FACTORISATION DOMAINS

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 𝒪q (GLn) and 𝒪q (SLn).

Fields of fractions of some SL q (2)-motion algebras

We construct new examples of iterated skew polynomial extensions. It is proved that quantum Gel’fand-Kirillov (GK) conjecture holds for this algebras. For one of the examples classical GK conjecture fails, but the quantum one is true.

A note on GK dimension of skew polynomial extensions

A note on GK dimension of skew polynomial extensions

Skew polynomial extensions of commutative Noetherian Jacobson rings

The Jacobson condition (i.e., that all prime ideals are semiprimitive) is proved to pass from a commutative noetherian ring R to a skew polynomial ring R [ y ; τ , δ ] R[y;\tau ,\delta ] , assuming only that τ \tau is an automorphism.