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An unrestricted notion of the finite factorization property

Jonathan Du, Felix Gotti

TL;DR

This work introduces and investigates the unrestricted finite factorization property (U-FF), a natural relaxation of the FF property that requires every atomic element to have finitely many factorizations, equivalently making the atomic submonoid an $FFM$. The authors position U-FF between the classical IDF and FF properties, prove that IDF implies U-FF but not vice versa, and analyze the behavior of U-FF under standard ring-theoretic constructions, notably the $D+M$ and polynomial extension frameworks. A key contribution is a precise transfer criterion for the U-FF property along D+M pullbacks, showing dependence on whether the maximal ideal contains irreducibles and on the finiteness of the unit-group quotient $K^ imes/k^ imes$. The paper also establishes that nearly atomic IDF domains are FFs and provides explicit examples where U-FF behaves differently from FF, including a domain with U-FF whose polynomial ring is not U-FF, highlighting the nuanced boundary between atomic and unrestricted arithmetic in integral domains.

Abstract

A nonzero element of an integral domain (or commutative cancellative monoid) is called atomic if it can be written as a finite product of irreducible elements (also called atoms). In this paper, we introduce and investigate an unrestricted version of the finite factorization property, extending the work on unrestricted UFDs carried out by Coykendall and Zafrullah who first studied unrestricted. An integral domain is said to have the unrestricted finite factorization (U-FF) property if every atomic element has only finitely many factorizations, or equivalently, if its atomic subring is a finite factorization domain (FFD). We position the property U-FF within the hierarchy of classical finiteness conditions, showing that every IDF domain is U-FF but not conversely, and we analyze its behavior under standard constructions. In particular, we determine necessary and sufficient conditions for the U-FF property to ascend along $D+M$ extensions, prove that nearly atomic IDF domains are FFDs, and construct an explicit example of an integral domain with the U-FF property whose polynomial ring is not U-FF. These results demonstrate that the U-FF property behaves analogously to the IDF property, while providing a finer interpolation between the IDF and the FF conditions.

An unrestricted notion of the finite factorization property

TL;DR

This work introduces and investigates the unrestricted finite factorization property (U-FF), a natural relaxation of the FF property that requires every atomic element to have finitely many factorizations, equivalently making the atomic submonoid an . The authors position U-FF between the classical IDF and FF properties, prove that IDF implies U-FF but not vice versa, and analyze the behavior of U-FF under standard ring-theoretic constructions, notably the and polynomial extension frameworks. A key contribution is a precise transfer criterion for the U-FF property along D+M pullbacks, showing dependence on whether the maximal ideal contains irreducibles and on the finiteness of the unit-group quotient . The paper also establishes that nearly atomic IDF domains are FFs and provides explicit examples where U-FF behaves differently from FF, including a domain with U-FF whose polynomial ring is not U-FF, highlighting the nuanced boundary between atomic and unrestricted arithmetic in integral domains.

Abstract

A nonzero element of an integral domain (or commutative cancellative monoid) is called atomic if it can be written as a finite product of irreducible elements (also called atoms). In this paper, we introduce and investigate an unrestricted version of the finite factorization property, extending the work on unrestricted UFDs carried out by Coykendall and Zafrullah who first studied unrestricted. An integral domain is said to have the unrestricted finite factorization (U-FF) property if every atomic element has only finitely many factorizations, or equivalently, if its atomic subring is a finite factorization domain (FFD). We position the property U-FF within the hierarchy of classical finiteness conditions, showing that every IDF domain is U-FF but not conversely, and we analyze its behavior under standard constructions. In particular, we determine necessary and sufficient conditions for the U-FF property to ascend along extensions, prove that nearly atomic IDF domains are FFDs, and construct an explicit example of an integral domain with the U-FF property whose polynomial ring is not U-FF. These results demonstrate that the U-FF property behaves analogously to the IDF property, while providing a finer interpolation between the IDF and the FF conditions.

Paper Structure

This paper contains 24 sections, 7 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algebraic Number Theory and Arithmetic Geometry
  3. Commutative Algebra and Module Theory
  4. Additive Combinatorics and Zero-Sum Theory
  5. Algebraic Geometry and Invariant Theory
  6. Computational and Algorithmic Applications
  7. The Bounded and Finite Factorization Properties
  8. The Unrestricted Finite Factorization Property
  9. Outline of the Paper
  10. Background
  11. General Notation
  12. Commutative Monoids
  13. Divisibility and Atomicity
  14. Factorizations
  15. Integral Domains and Monoid Domains
  16. ...and 9 more sections

Key Result

Theorem 2.2

fHK92 A monoid is an FFM if and only if it is an atomic IDF monoid.

Figures (3)

  • Figure 1: The implications in the diagram show the inclusions among subclasses of atomic monoids (AT stands for the class of atomic monoids). The (red) marked arrows emphasize that none of the shown implications are reversible.
  • Figure 2: The implications in the diagram show three properties implied by the FF property: the BF property, the IDF property, and and the MCD-finite property. The (red) marked arrows emphasize that none of the shown implications are reversible.
  • Figure 3: The diagram shows all the implications among the discussed properties generalizing the FF property. The (red) marked arrows emphasize that none of the shown implications are reversible.

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 3.1
  • Example 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • ...and 18 more