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On noncommutative finite factorization domains

Authors: Jason P. Bell; Albert Heinle; Viktor Levandovskyy;

On noncommutative finite factorization domains

Abstract

A domain $R$ is said to have the finite factorization property if every nonzero non-unit element of $R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $k$ be an algebraically closed field and let $A$ be a $k$-algebra. We show that if $A$ has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite then $A$ is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces and shift algebras. This provides a termination criterion for factorization procedures over these algebras. In addition, we give explicit upper bounds on the number of distinct factorizations of an element in terms of data from the filtration.

Related Organizations
Subjects by Vocabulary

Microsoft Academic Graph classification: Pure mathematics Noncommutative geometry Factorization Lie algebra Domain (ring theory) Filtration (mathematics) Component (group theory) Algebraically closed field Element (category theory) Mathematics

Keywords

Applied Mathematics, General Mathematics, Mathematics - Rings and Algebras, Rings and Algebras (math.RA), FOS: Mathematics

22 references, page 1 of 3

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D. Anderson and Bernadette Mullins. Finite factorization domains. Proceedings of the American Mathematical Society, 124(2):389-396, 1996.

Serguei P. Tsarev. An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator. In Proceedings of the 1996 international symposium on Symbolic and algebraic computation, pages 226-231. ACM, 1996.

Daniel Anderson. Factorization in integral domains, volume 189. CRC Press, 1997.

P. M. Cohn. Noncommutative unique factorization domains. Transactions of the American Mathematical Society, 109(2):313-331, 1963.

Nicholas R. Baeth and Daniel Smertnig. Factorization theory in noncommutative settings. 2014. URL http://arxiv.org/abs/1402.4397v1.

John C. McConnell and James Christopher Robson. Noncommutative noetherian rings, volume 30. American Mathematical Soc., 2001.

J. Apel. Gr¨obnerbasen in nichtkommutativen Algebren und ihre Anwendung. Dissertation, Universita¨t Leipzig, 1988.

V. Levandovskyy and H. Sch¨onemann. Plural - a computer algebra system for noncommutative polynomial algebras. In Proc. of the International Symposium on Symbolic and Algebraic Computation (ISSAC'03), pages 176 - 183. ACM Press, 2003. URL http://doi.acm.org/10.1145/860854.860895. [OpenAIRE]

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  • citations
    This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    15
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Top 10%
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Top 10%
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Top 10%
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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
15
Top 10%
Top 10%
Top 10%
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  • Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)
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