On Noncommutative Finite Factorization Domains
Abstract
A domain $R$ is said to have the finite factorization property if every nonzero nonunit 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 finitedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.6178
 Bibcode:
 2014arXiv1410.6178B
 Keywords:

 Mathematics  Rings and Algebras
 EPrint:
 doi:10.1090/tran/6727