MCD-finite Domains and Ascent of IDF Property in Polynomial Extensions

TL;DR

This paper introduces MCD-finite domains and characterizes when the IDF property ascends to polynomial extensions, providing new insights and counterexamples in the theory of integral domains.

Contribution

It defines MCD-finite domains and proves that polynomial extension preserves the IDF property if and only if the base domain is both IDF and MCD-finite, unifying previous conditions.

Findings

D[X] is IDF iff D is IDF and MCD-finite

Established new counterexamples for IDF ascent

Unified conditions for IDF property in polynomial extensions

Abstract

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCD-finite domains, and show that for any domain $D$, $D[X]$ is an IDF domain if and only if $D$ is both IDF and MCD-finite. This result entails all the previously known sufficient conditions for the ascent of the IDF property. Our new characterization of polynomial domains with the IDF property enables us to use a different construction and build another counterexample which strengthen the previously known result on this matter.