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=N0: 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 exhibit 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 find a torsion-free atomic monoid M 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 Z2[x;M] is not atomic.