MCD-finite Domains and Ascent of 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.

Integral domains and the IDF property

“…A common divisor d ∈ D of a nonempty subset S of D * is called a maximal common divisor if gcd S/d = 1. Following [18], we say that an integral domain D is MCD-finite if every finite subset of D * has only finitely many maximal common divisors up to associates.…”

“…It is clear that every GCD-domain is an MCD-finite domain, and it is not hard to verify that every pre-Schreier domain is an MCD-finite domain. More recently, Eftekhari and Khorsandi [18,Theorem 2.1] proved that the IDF property ascends in the class of MCD-finite domains. and r ∈ R, we set S ≥r = {s ∈ S | s ≥ r} and S >r = {s ∈ S | s > r}.…”

On properties of square-free elements in commutative cancellative monoids

“…We can also determine all the square-free elements of T : Using Proposition 8.6 we easily verify that L + xF [x] fulfills (i) -(vi). If F and L are finite fields and it is a proper extension, then L + xF [x] is a non-factorial ACCP domain (see [2], [9]). 9 The number of square-free elements of a reduced monoid…”

“…Let us extract possible combinations of (i) -(iv) for: atomic, ACCP, decomposition and GCD-monoids. We have: If F and L are finite fields and it is a proper extension, then L + xF [x] is a non-factorial ACCP domain (see [2], [9]). 9 The number of square-free elements of a reduced monoid…”

“…We have: If F and L are finite fields and it is a proper extension, then L + xF [x] is a non-factorial ACCP domain (see [2], [9]). 9 The number of square-free elements of a reduced monoid…”