The Relation between Almost Noetherian Module, Almost Finitely Generated Module and T-Noetherian Module

Abstract

In this paper we study the relation between almost Noetherian modules, almost finitely generated (a.f.g.) modules, and T-Noetherian modules. We show that if R’ = {r ∈ R|rM ≠ M} and M is an almost Noetherian (a.f.g. resp.) R-module, then M is an (R’)-Noetherian module. We also obtain that for any multiplicatively closed subset T of a ring R and R’ = {r ∈ R|rM ≠ M}, if M is an almost Noetherian (a.f.g. resp.) R-module and , then M is (T ∩ R’)-Noetherian. Moreover, we show that if M is an almost Noetherian (a.f.g. resp.) R-module and , then M is an T-Noetherian module for every multiplicatively closed set T ⊆ R. Finally, we apply the results of this paper on the structure of Generalized Power Series Module (GPSM) M[[S]].