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Divide and Transfer: Non-Unique Factorizations Beyond Commutativity

Daniel Smertnig

TL;DR

The work surveys divisor and transfer homomorphisms as unifying tools for non-unique factorizations across commutative and noncommutative algebra. In commutative Dedekind domains, it recasts element factorization through principal ideals and class groups using zero-sum sequences, linking factorization lengths to the structure of the class group. It then extends these ideas to noncommutative Dedekind prime rings via composition factors of finite-length modules, establishing a transfer to zero-sum sequence monoids under bounded Hermite hypotheses, and illustrates half-factoriality with concrete examples. Finally, it generalizes to ideals in hereditary noetherian prime rings using Rump–Yang divisors and a diagrammatic calculus, showing how ideal multiplication can be modeled by divisor composition and highlighting rich open problems in length sets and noncommutative factorization theory.

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.

Divide and Transfer: Non-Unique Factorizations Beyond Commutativity

TL;DR

The work surveys divisor and transfer homomorphisms as unifying tools for non-unique factorizations across commutative and noncommutative algebra. In commutative Dedekind domains, it recasts element factorization through principal ideals and class groups using zero-sum sequences, linking factorization lengths to the structure of the class group. It then extends these ideas to noncommutative Dedekind prime rings via composition factors of finite-length modules, establishing a transfer to zero-sum sequence monoids under bounded Hermite hypotheses, and illustrates half-factoriality with concrete examples. Finally, it generalizes to ideals in hereditary noetherian prime rings using Rump–Yang divisors and a diagrammatic calculus, showing how ideal multiplication can be modeled by divisor composition and highlighting rich open problems in length sets and noncommutative factorization theory.

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.
Paper Structure (6 sections, 18 theorems, 26 equations, 2 figures)

This paper contains 6 sections, 18 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting out
  3. Familiar Landscapes: Commutative Dedekind Domains
  4. The Noncommutative Frontier: Dedekind Prime Rings
  5. Uncharted Peaks: Ideals in HNP rings
  6. The Horizon: Open Problems

Key Result

Proposition 2

The map $\partial \colon R^\bullet \to \mathbb{N}_0^{(\mathop{\mathrm{Max}}\nolimits(R))}$ is a divisor homomorphism.

Figures (2)

  • Figure 1: Visualization of the divisors corresponding to the idempotent maximal ideals $Q_1$, $Q_2$, $Q_3$, the maximal invertible ideal $J(T_2)$ with $\partial(J(T_2)) = Q_1 + Q_2 + Q_3$, and the more complicated divisor $7Q_1 + 6Q_2 + 8Q_3$. (Colors for easier readability.)
  • Figure 2: The divisor $D=3Q_1+2 Q_2 + Q_3$ factored in two different ways, as $D=Q_1 \circ Q_2 \circ Q_3$ and $D=Q_2 \circ Q_1 \circ Q_3 \circ Q_2 \circ Q_3$.

Theorems & Definitions (39)

  • Definition 1
  • Proposition 2
  • proof
  • Definition 3
  • Lemma 4
  • proof
  • Definition 5
  • Definition 6
  • Theorem 7
  • proof
  • ...and 29 more