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

References in corpus (3)

• A construction of integer-valued polynomials with prescribed sets of lengths of factorizations
• Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields
• Factorization of integer-valued polynomials with square-free denominator

Cited by in corpus (4)

• Absolute irreducibility of the binomial polynomials
• Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains
• Primes and absolutely or non-absolutely irreducible elements in atomic domains
• Irreducible integer-valued polynomials with prescribed minimal power that factors non-uniquely