Elasticity in polynomial-type extensions

TL;DR

This paper investigates the relationship between the elasticity of atomic integral domains and their polynomial rings, identifying conditions under which the polynomial ring's elasticity equals 1, thus linking factorization properties.

Contribution

It characterizes integral domains R for which the polynomial ring R[x] has elasticity 1 based on irreducibility conditions of polynomials in R[x].

Findings

If R has at least one atom, R[x] has elasticity 1 when every nonconstant irreducible polynomial in R[x] remains irreducible in K[x].

The paper determines the specific integral domains R satisfying this irreducibility condition.

Provides criteria connecting the elasticity of R and R[x] in the context of atomic integral domains.

Abstract

The elasticity of an atomic integral domain is, in some sense, a measure of how far the domain is from being a unique factorization domain (or, more properly, a half-factorial domain). We consider the relationship between the elasticity of a domain, R, and the elasticity of its polynomial ring R[x]. For example, if R has at least one atom, a sufficient condition for the polynomial ring R[x] to have elasticity 1 is that every nonconstant irreducible polynomial f in R[x] be irreducible in K[x]. We will determine the integral domains R whose polynomial rings satisfy this condition.