Divide and Transfer: Non-Unique Factorizations Beyond Commutativity

TL;DR

This paper explores the extension of divisor and transfer homomorphisms from commutative to noncommutative rings, providing new insights into non-unique factorizations using module theory and diagrammatic calculus.

Contribution

It introduces a framework for understanding non-unique factorizations in noncommutative rings through adapted divisor theories and homomorphisms, expanding classical concepts.

Findings

Application of divisor homomorphisms to noncommutative Dedekind prime rings

Development of a diagrammatic calculus for divisors in hereditary noetherian prime rings

Extension of combinatorial models to noncommutative algebraic structures

Abstract

Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we first explore these notions in the classical setting of commutative Dedekind domains, where monoids of zero-sum sequences appear as a natural combinatorial model. We then adapt these ideas to the setting of noncommutative Dedekind prime rings using module-theoretic methods. Going a step further, we discuss Rump and Yang's recent divisor theory for ideals in hereditary noetherian prime rings, where divisors can be visualized in a diagrammatic calculus.