Factorization theory: From commutative to noncommutative settings
By Nicholas R. Baeth, Daniel Smertnig. 2014
73 citations recorded by the source corpus.
Abstract We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the ω-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of n × n upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/20f4b58905a70acc76cd05e0dfaeb0c986bc1c7c
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In Algebra and Number Theory
In Geometry and Topology
Citations in the corpus
- Power monoids: A bridge between factorization theory and arithmetic combinatorics (2017, background): “In § 2, we first extend a few fundamental aspects of the classical theory of non-unique factorization to a non-commutative and non-cancellative setting, in the same spirit of Baeth and Smertnig’s work on the factorization theory of non-commutative, cancellative monoids [5] (see Remarks 2.”