On the atomicity of monoid algebras

[摘要] 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 exhibit for any infinite cardinal kappa a torsion-free atomic monoid M of rank kappa 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 Z(2) [x; M] is not atomic. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-01

[关键词] Monoid algebras;Atomicity;Atomic algebras;Atomic Puiseux monoids