Factorizations of Elements in Noncommutative Rings: A Survey

Abstract

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.

50 pages, comments welcome

References in corpus (4)

• Quantum cluster algebras and quantum nilpotent algebras
• On Ritt's polynomial decomposition theorems
• Factorizations in bounded hereditary Noetherian prime rings
• From the Fundamental Theorem of Algebra to Kempe's Universality Theorem

Cited by in corpus (13)

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• Factorizations in bounded hereditary Noetherian prime rings
• Sets of Arithmetical Invariants in Transfer Krull Monoids
• Arithmetical invariants of local quaternion orders
• Surfaces containing two circles through each point
• Surfaces containing two circles through each point and Pythagorean 6-tuples
• A characterization of length-factorial Krull monoids
• On the Factorization of Non-Commutative Polynomials (in Free Associative Algebras)
• A Factorization Theory for some Free Fields
• Factorization in monoids and rings
• Factorization Theory in Commutative Monoids
• Factorization in Finitely-Presented Monoids
• Elasticities of Orders in Central Simple Algebras