Envelopes and Weakly Radicals of Submodules

TL;DR

This paper introduces a method to compute envelopes of submodules in finitely generated modules over Noetherian rings, explores the relationship between weakly prime submodules and their envelopes, and characterizes the weakly radical of submodules.

Contribution

It provides a new computational approach for envelopes and clarifies the connection between weakly prime submodules and their envelopes, advancing the understanding of weakly radicals.

Findings

A method for computing the envelope of a submodule is developed.

Relations between weakly prime submodules and their envelopes are established.

The weakly radical of a submodule is characterized using these relations.

Abstract

Let $N$ be a submodule of a finitely generated module $M$ over a Noetherian ring. A method for the computation of the submodule generated by the envelope of $N$ is given. The relations between weakly prime submodules and their envelopes are investigated. Using these relations, a description of the weakly radical of a submodule is obtained. The results are illustrated by examples.