Some variants of ascending and descending chain conditions

Abstract

We introduce the idea of nei-Noetherian and nei-Artinian modules and rings. We discuss some examples of nei-Noetherian (nei-Artinian) modules and study several properties of them. We characterize these modules such that an R module M is nei-Noetherian (nei-Artinian) if and only if every non essential submodule of M is iso-Noetherian (iso-Artinian) if and only if every proper closed submodule of M is iso-Noetherian (iso-Artinian). We further characterize semiprime right semihereditary nei-Noetherian rings. Finally, we discuss some other variants of ascending and descending chain conditions on the classes of non summands.

Keywords:

• ACC
• DCC
• ei-Artinian modules
• ei-Noetherian modules
• nei-Noetherian modules
• nei-Artinian modules

2020 Mathematics Subject Classification:

• 16P20
• 16P40
• 16P60
• 16P70
• 16E05
• 16E10

Acknowledgement

The authors wish to thank referees for their careful reading, pointing out some errors and comments to improve this paper. The second author is grateful to the CMP Degree College for their support.