On the atomicity of monoid algebras

Abstract

Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x; M ] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M = N 0 : he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F [x; M ] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we find for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x; M ] is not atomic for any integral domain R. Then for every n ≥ 2 and for each field F of finite characteristic we exhibit a torsion-free atomic monoid of rank n such that F [x; M ] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z 2 [x; M ] is not atomic.

Divisibility in rings of integer-valued polynomials

“…The atomicity of additive submonoids of Q ≥0 has been systematically investigated during the last few years (see the recent survey [13] and references therein). As we will confirm here, these monoids are effective to find counterexamples in commutative ring theory (see also [20]).…”

Atomicity of Positive Monoids

“…Consider now the monoid M = N 0 [α]. It follows from [17,Proposition 3.2] that the rank of M equals the degree of m(x), that is, rank M = n. Because α > 1, the monoid M is an FFM by virtue of Theorem 6.1. Finally, let us show that M is not a UFM.…”

A characterization of finite factorization positive monoids

“…For example, Grams [17] used Puiseux monoids (i.e., additive submonoids of Q ≥0 ) to refute Cohn's assertion that every atomic integral domain satisfies the ascending chain condition on principal ideals ([7, Proposition 1.1]). More recently, Bras-Amorós [4] highlighted connections between positive monoids and music theory, while Coykendall and Gotti [9] employed Puiseux monoids to tackle a question posed by Gilmer almost four decades ago in [14, page 189]. The aim of the present article is to study the positive monoids that satisfy the finite factorization property.…”