Nilpotent elements control the structure of a module

TL;DR

This paper explores how nilpotent elements influence the structure of modules, showing that their presence can hinder desirable properties and providing examples to distinguish various types of prime modules.

Contribution

It introduces a general form of an example that differentiates between several classes of prime modules and their properties related to nilpotency.

Findings

Nilpotent elements inhibit modules from having good structural properties.

Reduced modules can be expressed as sums of prime modules.

Examples distinguish different classes of prime modules based on nilpotency and radical properties.

Abstract

A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula.