Some Extensions of Endo-Noetherian Rings

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 $\mathrm{\Lambda }:=\{f\in R[[S,\omega ]]:f(1)\in T\}$ 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.