Noetherian rings of composite generalized power series

Abstract

Let A⊆BA\subseteq B be an extension of commutative rings with identity, (S,≤)\left(S,\le ) a nonzero strictly ordered monoid, and S*=S\{0}{S}^{* }\left=S\backslash \left\{0\right\}. Let A+〚BS*,≤〛={f∈〚BS,≤〛∣f(0)∈A}A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] =\{f\in [\kern-2pt[ {B}^{S,\le }]\kern-2pt] \hspace{0.33em}| \hspace{0.33em}f\left(0)\in A\}. In this study, we determine when the ring A+〚BS*,≤〛A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when SS is a strict monoid, if A+〚BS*,≤〛A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then AA is a Noetherian ring, BB is a finitely generated AA-module, and SS is finitely generated. We also show that if BB is a finitely generated AA-module over a Noetherian ring AA and (S,≤)\left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A+〚BS*,≤〛A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.

Keywords

• generalized power series ring
• noetherian ring
• strictly ordered monoid
• 06f05
• 13a15
• 13e05
• 13f25