[Paper Review] A new simple example of an atomic domain which is not ACCP

Hamid Kulosman|arXiv (Cornell University)|May 25, 2018

Rings, Modules, and Algebras4 references3 citations

TL;DR

This paper presents a new, simple example of an atomic domain that does not satisfy the ascending chain condition on principal ideals (ACCP) by constructing a monoid domain $F[X;M]$, where $F$ is a field and $M$ is a submonoid of the nonnegative rationals generated by $\frac{1}{p_i p_{i+2}}$ for consecutive primes $p_i$. The domain is atomic because all generators are irreducible, but the monoid $M$ fails ACCP due to an infinite strictly increasing chain of principal ideals, which lifts to the domain via Proposition 3.1.

ABSTRACT

We give a new simple example of an atomic domain which is not ACCP. Our example is a monoid domain $F[X;M]$, where $F$ is a field and $M$ is a submonoid of the additive monoid of nonnegative rational numbers.

Motivation & Objective

• To provide a new, simple example of an atomic domain that is not ACCP, addressing a long-standing gap in factorization theory.
• To demonstrate that atomicity does not imply ACCP in integral domains, despite the converse being classically known.
• To construct such an example using a monoid domain $F[X;M]$ with $M$ a submonoid of $\mathbb{Q}_{+}$, enabling algebraic and combinatorial control over factorization properties.
• To clarify the distinction between atomicity and ACCP in monoid domains by explicitly analyzing the monoid structure and its ideal chain behavior.

Proposed method

• Define the monoid $M = \langle \frac{1}{p_i p_{i+2}} \mid i \geq 1 \rangle$ under addition, where $p_i$ are consecutive primes.
• Prove that each generator $\frac{1}{p_i p_{i+2}}$ is an atom in $M$ by showing it cannot be written as a sum of smaller positive elements in $M$, using denominator analysis.
• Construct an infinite strictly increasing chain of principal ideals in $M$: $\left(\frac{1}{p_i} + \frac{1}{p_{i+1}}\right) \subset \left(\frac{1}{p_{i+1}} + \frac{1}{p_{i+2}}\right)$, proving $M$ is not ACCP.
• Use Proposition 3.1 to transfer the ACCP failure from $M$ to the monoid domain $F[X;M]$, showing $F[X;M]$ is not ACCP.
• Prove $F[X;M]$ is atomic by contradiction: assuming an element factors infinitely, analyze the largest exponent in reduced form and show it leads to a denominator contradiction involving large primes.
• Leverage the fact that only finitely many copies of any atom can appear in a sum without altering the denominator structure, ensuring finite factorization.

Experimental results

Research questions

• RQ1 Can a simple, explicit example of an atomic domain that is not ACCP be constructed using monoid domains over submonoids of $\mathbb{Q}_{+}$?
• RQ2 Is there a monoid domain $F[X;M]$ that is atomic but fails ACCP, with $M$ a submonoid of the nonnegative rationals?
• RQ3 Does the atomicity of $M$ imply the atomicity of $F[X;M]$? (The paper does not resolve this in general, but shows it holds in this example.)
• RQ4 Can the failure of ACCP be demonstrated via an explicit infinite chain of principal ideals in the monoid $M$?
• RQ5 What structural properties of $M$ ensure that $F[X;M]$ remains atomic despite $M$ not being ACCP?

Key findings

• The monoid $M = \langle \frac{1}{p_i p_{i+2}} \mid i \geq 1 \rangle$ is atomic, as each generator $\frac{1}{p_i p_{i+2}}$ is irreducible due to denominator constraints in sums.
• The monoid $M$ fails ACCP because the sequence $\left(\frac{1}{p_i} + \frac{1}{p_{i+1}}\right)_{i \geq 1}$ forms an infinite strictly increasing chain of principal ideals.
• The monoid domain $F[X;M]$ is not ACCP, as established by Proposition 3.1, which links the ACCP property of $F[X;M]$ to that of $M$.
• The monoid domain $F[X;M]$ is atomic, proven by contradiction: an infinite factorization would require a denominator with a prime $p_j$ not present in the original exponent, leading to contradiction.
• The construction is simple and explicit, using only rational numbers and prime indices, making it accessible for further study in factorization theory.
• The example illustrates that atomicity does not imply ACCP, even in well-behaved monoid domains, and provides a new, elementary instance of this phenomenon.