$\tau_I$-Elasticity for quotients of order four

TL;DR

This paper investigates the minimal quotient of a UFD where elements can have atomic $ au_I$-factorizations of different lengths, revealing non-uniqueness in factorizations within certain algebraic structures.

Contribution

It identifies the smallest quotient of a UFD with an ideal where elements exhibit atomic factorizations of varying lengths, specifically for $ au_I$-factorizations.

Findings

Found a specific quotient of $ ext{Z}[x]$ with ideal $(2,x^2+x)$ where elements have factorizations of different lengths.

Constructed a sequence of elements with atomic $ au_I$-factorizations of length 2 and arbitrary larger lengths.

Demonstrated non-uniqueness of factorizations in the identified algebraic setting.

Abstract

For a commutative domain $R$ with nonzero identity and $I$ an ideal of $R$, we say $a=\lambda b_1 \cdots b_k$ is a $\tau_I$-factorization of $a$ if $\lambda \in R$ is a unit and $b_i \equiv b_j$(mod $I$) for all $1\leq i \leq j \leq k$. These factorizations are nonunique, and two factorizations of the same element may have different lengths. In this paper, we determine the smallest quotient $R/I$ where $R$ is a unique factorization domain, $I\subset R$ an ideal, and $R$ contains an element with atomic $\tau_I$-factorizations of different lengths. In fact, for $R=\mathbb{Z}[x]$ and $I = (2,x^2+x)$, we can find a sequence of elements $a_i$ that have an atomic $\tau_I$-factorization of length 2 and one of length $i$ for $i\in\mathbb{N}$.