Some Extensions of Endo-Noetherian Rings

TL;DR

This paper investigates how the left endo-Noetherian property transfers to various ring extensions, including power series rings, skew generalized power series rings, and amalgamated duplication rings, providing necessary and sufficient conditions.

Contribution

It establishes new criteria for the transfer of the left endo-Noetherian property to several classes of ring extensions, expanding understanding of this property in ring theory.

Findings

Transfer of left endo-Noetherian property to power series rings is characterized.

Subring conditions for endo-Noetherian property in skew generalized power series rings are established.

Results on endo-Noetherian property in amalgamated duplication rings are provided.

Abstract

In this article, we proceed on the transfer of the left endo-Noetherian property on certain ring extensions. We transfer of the right (left) endo-Noetherian property to the right (left) quotient rings. For a subring $T$ of $R$ and a finite set of indeterminates $X$, we prove that $T + XR[[X]]$ is left endo-Noetherian if and only if $R[[X]]$ is left endo-Noetherian. In addition, we prove that the subring $\Lambda :=\{ f \in R[[S,\omega ]]: f(1) \in T \}$ of the skew generalized power series ring $R[[S, \omega]]$ is left endo-Noetherian if and only if $R[[S, \omega]]$ is left endo-Noetherian. Also, we study the left endo-Noetherian property over the amalgamated duplication rings $R \bowtie I$ and $ R \bowtie ^f J$. Finally, we introduce additional results on left endo-Noetherian rings.