Noetherian rings of composite generalized power series

Abstract

Abstract Let A ⊆ B A\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 + 〚 B S * , ≤ 〛 = { f ∈ 〚 B S , ≤ 〛 ∣ 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 + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when S S is a strict monoid, if A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then A A is a Noetherian ring, B B is a finitely generated A A -module, and S S is finitely generated. We also show that if B B is a finitely generated A A -module over a Noetherian ring A A and ( S , ≤ ) \left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.