Factorization theory: From commutative to noncommutative settings

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 $\omega$-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 $n\times n$ upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.

45 pages

References in corpus (3)

• On Ritt's polynomial decomposition theorems
• Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic
• Algebraic and F-Independent sets in 2-firs

Cited by in corpus (30)

• Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics
• Factorizations in bounded hereditary Noetherian prime rings
• Geometry of free loci and factorization of noncommutative polynomials
• Structural properties of subadditive families with applications to factorization theory
• Systems of sets of lengths: Transfer Krull monoids versus weakly Krull monoids
• Factorizations in upper triangular matrices over information semialgebras
• Factorizations of Elements in Noncommutative Rings: A Survey
• The catenary degree of Krull monoids II
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• On the Arithmetic of Power Monoids and Sumsets in Cyclic Groups
• An Abstract Factorization Theorem and Some Applications
• A Characterization of class groups via sets of lengths {II}
• Long sets of lengths with maximal elasticity
• Factorization under Local Finiteness Conditions
• Arithmetical invariants of local quaternion orders
• Abstract Factorization Theorems with Applications to Idempotent Factorizations
• On products of k atoms II
• On the atomic structure of torsion-free monoids
• On the Factorization of Non-Commutative Polynomials (in Free Associative Algebras)
• On the Algebraic and Arithmetic structure of the monoid of Product-one sequences II
• On half-factoriality of transfer Krull monoids
• A Factorization Theory for some Free Fields
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra
• The Representation Theorem of Persistent Homology Revisited and Generalized
• Horner Systems: How to efficiently evaluate non-commutative polynomials (by matrices)
• Divisor sequences of atoms in Krull monoids
• Minimal relations and catenary degrees in Krull monoids
• Factorization in Finitely-Presented Monoids
• On transfer Krull monoids
• Factorization Theory in Commutative Monoids