Factorizations of Elements in Noncommutative Rings: A Survey

TL;DR

This survey reviews the theory of element factorizations in noncommutative rings, highlighting recent developments, invariants, and transfer results, inspired by the commutative non-unique factorization theory.

Contribution

It provides a comprehensive overview of factorization properties in noncommutative rings, including new insights into arithmetical invariants and transfer principles.

Findings

Characterization of factorizations in various noncommutative rings

Transfer results for arithmetical invariants in matrix and order rings

Connections between noncommutative and commutative factorization theories

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.