The nilpotency of the prime radical of a Goldie module

TL;DR

This paper proves that in certain Goldie modules, the intersection of all prime submodules is nilpotent, extending the classical result about prime radicals in Goldie rings.

Contribution

It extends the classical nilpotency of prime radicals from Goldie rings to a broader class of Goldie modules under specific conditions.

Findings

The intersection of all prime submodules in a Goldie module is nilpotent.

The result generalizes the nilpotency of prime radicals from rings to modules.

Conditions include the module being retractable and $M^{( ext{Lambda})}$-projective.

Abstract

With the notion of prime submodule defined by F. Raggi et.al. we prove that the intersection of all prime submodules of a Goldie module $M$, is a nilpotent submodule provided that $M$ is retractable and $M^{(\Lambda)}$-projective for every index set $\Lambda$. This extends the well known fact that in a left Goldie ring, the prime radical is nilpotent.