Non-absolutely irreducible elements in the ring of Integer-valued polynomials

TL;DR

This paper investigates the existence of elements in the ring of integer-valued polynomials that are irreducible but not absolutely irreducible, providing explicit constructions and generalizations.

Contribution

It introduces the concept of non-absolutely irreducible elements in the ring of integer-valued polynomials and offers explicit constructions and generalizations.

Findings

Constructed examples of non-absolutely irreducible elements in $ ext{Int}( ext{Z})$

Provided methods to generate such elements systematically

Extended constructions to broader classes of polynomials

Abstract

Let $R$ be a commutative ring with identity. An element $r \in R$ is said to be absolutely irreducible in $R$ if for all natural numbers $n>1$, $r^n$ has essentially only one factorization namely $r^n = r \cdots r$. If $r \in R$ is irreducible in $R$ but for some $n>1$, $r^n$ has other factorizations distinct from $r^n = r \cdots r$, then $r$ is called non-absolutely irreducible. In this paper, we construct non-absolutely irreducible elements in the ring $\text{Int}(\mathbb{Z}) = \{f\in \mathbb{Q}[x] \mid f(\mathbb{Z}) \subseteq \mathbb{Z}\}$ of integer-valued polynomials. We also give generalizations of these constructions.