Modules Satisfying the S-Noetherian Property and S-ACCR

Abstract

Let R be a commutative ring with unity, S a multiplicative subset of R, and M an R-module. In this article, we investigate S-Noetherian modules. We give an S-version of Eakin–Nagata–Formanek Theroem [7], in the case where S is finite. We prove that if M is an S-finite R-module and any increasing chain of extended submodules of M is S-stationary then M is S-Noetherian. In the second part of this article, we define S-accr modules. An R-module M is said to satisfy S-accr if any ascending chain of residuals of the form (N: B) ⊆ (N: B2) ⊆ (N: B3) ⊆ … is S-stationary where N is a submodule of M and B is a finitely generated ideal of R. We investigate the class of such modules M, and we generalize some known results of P. C. Lu ([5], [6]).