Maximal common divisors in monoid algebras

Abstract

We say that a commutative monoid $M$ has the MCD property if every nonempty finite subset of $M$ has a maximal common divisor (MCD), and we say that $M$ has the MCD-finite property if every nonempty finite subset of $M$ has only finitely many MCDs up to associates. It is well known that every monoid that satisfies the ascending chain condition on principal ideals is an MCD monoid, while every finite factorization monoid and every pre-Schreier monoid is an MCD-finite monoid. In this paper, we study the MCD and MCD-finite properties in the setting of monoid algebras. After identifying a new class of rank-$1$ torsion-free MCD monoids, we investigate the ascent of the MCD property to monoid algebras over fields, proving that if a pre-Schreier monoid has the MCD property then its monoid algebras over any field also have the MCD property. Then we prove that, unlike for the case of polynomial extensions, the property of being atomic does not ascend to monoid algebras over fields when restricted to the class of MCD monoids. In the second part of the paper, we first identify a new class of rank-$1$ torsion-free MCD-finite monoids. Then we establish the ascent of the MCD-finite property to polynomial extensions. We conclude the paper proving that the q-GCD property (i.e., the condition that every nonempty finite subset has at most one MCD), which is a condition stronger than the MCD-finite property, does not ascend to monoid algebras over fields even when restricted to the class of rank-$1$ torsion-free monoids.