On AB5* modules with Noetherian dimension

TL;DR

This paper investigates the Noetherian dimension of modules with specific properties, providing bounds, structure theorems, and classifications for modules in the AB5* category.

Contribution

It establishes bounds on Noetherian dimension for modules with finite spanning dimension or weakly atomic modules, and classifies alpha-short modules in the AB5* category.

Findings

Noetherian dimension of certain modules is bounded by alpha.

Structure theorem for alpha-short modules in AB5* category.

Classification of alpha-short modules in the AB5* category.

Abstract

In this paper, we study the Noetherian dimension of sum of certain modules. It is proved that for any module M which is an irredundant sum of submodules, each of which has Noetherian dimension less than alpha, if M has finite spanning dimension (fsd-module, for short) or it is a weakly atomic module, then Noetherian dimension M less than alpha. Here, by a weakly atomic module we mean a module M for which every proper non-small submodule N, has Noetherian dimension strictly less than that of M. Also, it is proved that if M is an AB5* module with Noetherian dimension and N_i is a family of submodules of M such that Noetherian dimension M over N_i, less than alpha, for each i, then Noetherian dimension M over intersection of N_is less than alpha. Using this, we give a structure theorem for alpha-short modules in the category of AB5* and finally, we classify alpha-short modules in this category.