# Counting submodules of a module over a noetherian commutative ring

## TL;DR

This paper provides a method to count submodules of modules over countable noetherian commutative rings and characterizes uniserial modules, offering structural insights into meager modules.

## Contribution

It introduces a counting technique for submodules and characterizes uniserial modules over such rings, along with describing meager modules structurally.

## Key findings

- Count of submodules for modules over countable noetherian rings
- Structural description of meager modules
- Characterization of uniserial modules

## Abstract

We count the number of submodules of an arbitrary module over a countable noetherian commutative ring. We give, along the way, a structural description of meager modules, which are defined as those that do not have the square of a simple module as subquotient. We deduce in particular a characterization of uniserial modules over commutative noetherian rings.