Some Extensions of Endo-Noetherian Rings

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 $Λ:=\{ f \in R[[S,ω]]: f(1) \in T \}$ of the skew generalized power series ring $R[[S, ω]]$ is left endo-Noetherian if and only if $R[[S, ω]]$ 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.