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# On noncommutative finite factorization domains

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# On noncommutative finite factorization domains

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.

- University of Waterloo Canada
- RWTH Aachen University Germany

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

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

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

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

###### 22 references, page 1 of 3

D. D. Anderson, David F. Anderson, and Muhammad Zafrullah. Factorization in integral domains. Journal of pure and applied algebra, 69(1):1-19, 1990. [OpenAIRE]

D. D. Anderson and David F. Anderson. Elasticity of factorizations in integral domains. Journal of pure and applied algebra, 80(3):217-235, 1992. [OpenAIRE]

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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- Funder: Natural Sciences and Engineering Research Council of Canada (NSERC)

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.

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