Ascent of atomicity from torsion-free commutative monoids to monoid algebras

Determine whether atomicity ascends from torsion-free commutative monoids to their monoid algebras: specifically, for every torsion-free commutative monoid M and every commutative ring R, ascertain whether the atomicity of M implies that the monoid algebra R[M] is atomic.

Background

Atomicity is a fundamental property in factorization theory, requiring every non-invertible element to factor as a finite sum of atoms. The paper focuses on the ascent of atomic and factorization properties from Puiseux monoids to their power monoids, noting broader contexts where ascent questions have been studied.

In particular, the authors reference the classical ascent question for monoid algebras (also called semigroup rings): whether atomicity of a torsion-free commutative monoid M passes to its monoid algebra R[M]. They note this problem’s historical origin and subsequent investigations, indicating ongoing interest and partial progress in related settings.