On seminoetherian rings and modules

Abstract

We say that a module M has enough noetherians if any nonzero submodule of M contains a nonzero noetherian submodule. We prove that the class of modules having enough noetherians is closed under submodules, essential extensions, direct sums, and module extensions. Characterizations of these modules are provided. We call a module M seminoetherian if any nonzero factor module of M contains a nonzero noetherian submodule. We show that a module M is seminoetherian if and only if the $R[X]$-module $M[X]$ is seminoetherian if and only if M has Gabriel dimension and for every $\mathfrak{p}\in \mathit{Supp}(M),R/\mathfrak{p}$ is noetherian.

Keywords:

• Conoetherian ideals
• Gabriel dimension
• modules (rings) having enough noetherians
• seminoetherian modules (rings)

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C13
• 13B25
• 16D10
• 16D80

Acknowledgments

The authors would like to thank the referee for the careful reading of the manuscript and the useful comments which improved this paper.