Modules Satisfying the S-Noetherian Property and S-ACCR

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 Lee, S. C., Kim, S., Chung, S. C. (2005). Ideals and submodules of multiplications modules. J. Korean Math. Soc. 42(5):933–948.[Crossref], [Web of Science ®] , [Google Scholar]], 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 Lu, C. P. (1988). Modules satisfying ACC on a certain type of colons. Pacific J. Math. 131(2):303–318.[Crossref], [Web of Science ®] , [Google Scholar]], [6 Lu, C. P. (1993). Modules and rings satisfying ACCR. Proc of The AMS 117(1):5–10.[Crossref], [Web of Science ®] , [Google Scholar]]).