Non-Commutative Unique Factorisation Rings

Chatters, A. W.; Jordan, David A.

doi:10.1112/jlms/s2-33.1.22

Cited by 61 publications

(81 citation statements)

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“…The one obvious exception is localisability, and there is no hope of proving this because we did not know enough about the ring R/pR. We will show in Section 4 that pR is localisable if R satisfies a polynomial identity, and the Noetherian case was dealt with in the proof of Theorem 2.1 of [6].…”

Section: General Theorymentioning

confidence: 99%

“…(unique factorisation domain) if R is an integral domain and every non-zero prime ideal contains a completely prime element. These definitions were introduced for Noetherian rings in [5] and [6]. We shall repeatedly use the fact that if x is a normal element of R and P is a prime ideal of R, then either x e P or xeC(P); this is because (xR + P)/P=(Rx + P)/P is a twosided ideal of the prime ring R/P, and an ideal of a prime ring is either 0 or its left and right annihilators are 0.…”

Section: Definitions Notation and Preliminariesmentioning

confidence: 99%

“…The concept of a unique factorisation domain in commutative algebra was extended in two different ways to non-commutative rings in [5] and [6]. However, these generalisations were only studied in the context of Noetherian rings.…”

Section: Introductionmentioning

confidence: 99%

“…However, these generalisations were only studied in the context of Noetherian rings. Even here there is a surprisingly rich supply of genuinely non-commutative examples, including trace rings of generic matrix rings and many twisted polynomial rings, group rings and universal enveloping algebras (see [1], [5], [6], [13]). The aim of this paper is to develop a theory of non-commutative U.F.R.S (unique factorisation rings) without the Noetherian condition.…”

Section: Introductionmentioning

confidence: 99%

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Unique factorisation rings

Chatters

Gilchrist

Wilson

1992

Proceedings of the Edinburgh Mathematical Society

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Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

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Section: General Theorymentioning

confidence: 99%

Section: Definitions Notation and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unique factorisation rings

Chatters

Gilchrist

Wilson

1992

Proceedings of the Edinburgh Mathematical Society

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“…Recall from [7] that a noetherian domain R is called a unique factorization domain (UFD) if every non-zero prime ideal of R contains a non-zero completely prime ideal of the form pR where p is normal in R. The following is immediate from Theorem 3.13.…”

Section: Proof (I)mentioning

confidence: 91%

Connected quantized Weyl algebras and quantum cluster algebras

Fish

Jordan

2018

Journal of Pure and Applied Algebra

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Abstract. For an algebraically closed field K, we investigate a class of noncommutative K-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators {x 1 , . . . , x n } such that each pair satisfies a relation of the form x i x j = q ij x j x i + r ij , where q ij ∈ K * and r ij ∈ K, with, in some sense, sufficiently many pairs for which r ij = 0. For such an algebra it turns out that there is a single parameter q such that each q ij = q ±1 . Assuming that q = ±1, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let n ≥ 3 be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type A n−1 as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver P (1) n+1 identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in n + 2 variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of A n−1 and P (1) n+1 are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element. Key words and phrases. skew polynomial ring, quantized Weyl algebra, quantum cluster algebra, Poisson algebra.Some of the results in this paper appear in the University of Sheffield PhD thesis of the first author, supported by the Engineering and Physical Sciences Research Council of the UK. We thank Vladimir Dotsenko for his helpful comments at an early stage of this project, the referee for many helpful suggestions of improvements in the exposition and Bach V Nguyen for pointing out some errors in our original manuscript.

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Semigroup Algebras and Noetherian Maximal Orders

Jespers

Okniński

2001

Journal of Algebra

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w xIn this paper we describe when a monoid algebra K S is a noetherian PI domain which is a maximal order. Our work relies on the study of the height one w x primes of K S and of the minimal primes of the monoid S and leads to a characterization purely in terms of S. It turns out that the primes P intersecting S plays a crucial role, and therefore we reduce the problem to certain ''local'' monoids S , that is, monoids with only one minimal prime. However, we illustrate P by examples that such monoids and their algebras are much more complicated than the discrete valuation rings arising in the commutative case. Our work is based on the results of Brown concerning the group ring case and of Anderson and Chouinard on commutative monoid rings. We also rely on the description of noetherian monoid algebras obtained earlier by the authors.

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Non-Commutative Unique Factorisation Rings

Cited by 61 publications

References 3 publications

Unique factorisation rings

Unique factorisation rings

Connected quantized Weyl algebras and quantum cluster algebras

Semigroup Algebras and Noetherian Maximal Orders

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