The Sufficient Conditions for R[X]-module M[X] to be S[X]-Noetherian

Abstract

In this paper, we obtain the necessary and sufficient conditions on a ring $R$, a multiplicative set $S\subseteq R$, and an $R$-module $M$ such that the polynomial module, the Laurent polynomial module, the power series module, and the Laurent series module are $S$-Noetherian. Furthermore, we also obtain the sufficient conditions for these modules to be $S[X]$-Noetherian, $S[X,X^{-1}]$-Noetherian, $S[[X]]$-Noetherian, and $S[[X,X^{-1}]]$-Noetherian, respectively.