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.

Elasticities of Orders in Central Simple Algebras

“…A factorization of an element need not be unique in general, and in fact, two factorizations of the same element may have different lengths. We recall some important definitions as seen in [Sme16].…”

A characterization of length-factorial Krull monoids

“…Thus, every commutative Krull domain is a normalizing Krull ring. For examples and background on non-commutative (normalizing) Krull rings we refer to [53,10,45,1], and for background on factorizations in the non-commutative setting to [4,52]. In particular, normalizing Krull monoids are transfer Krull.…”

A Factorization Theory for Some Free Fields

“…Definition 2.18 (Similarity Unique Factorization Domains [23]). A domain R is called similarity factorial (or a similarity-UFD) if R is atomic and it satisfies the…”