S-Noetherian generalized power series rings

TL;DR

This paper investigates the transfer of the right S-Noetherian property to skew generalized power series rings, establishing conditions under which these rings inherit Noetherian properties from their base rings and monoids.

Contribution

It unifies and generalizes existing results on Noetherian properties in skew power series rings and provides new criteria for these properties to hold in generalized settings.

Findings

R[[M,w]] is left Noetherian iff R is left Noetherian and M is finitely generated.

Transfer of S-Noetherian property from R to R[[M,w]] under certain conditions.

Equivalence of right S-Noetherian property between R and skew polynomial rings R[x,a].

Abstract

Let R be a ring with identity, (M;\leq) a commutative positive strictly ordered monoid and w_m an automorphism for each m \in M . The skew generalized power series ring R[[M,w]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev Neumann Laurent series rings. If S\subset R is a multiplicative set, then R is called right S-Noetherian, if for each ideal I of R, Is \subseteq J\subseteq I for some s\in S and some finitely generated right ideal J . Unifying and generalizing a number of known results, we study transfers of S-Noetherian property to the ring R[[M,w]]. We also show that the ring R[[M,w]] is left Noetherian if and only if R is left Noetherian and M is finitely generated. Generalizing a result of Anderson and Dumitrescu, we show that,when S\subset R is a-anti-Archimedean multiplicative set with a an automorphism of R, then R is right S-Noetherian if and only if the skew polynomial ring R[x,a] is right S-Noetherian.