Modules with Non-Cyclic Socle and the Extension Property

TL;DR

This paper investigates the conditions under which modules with non-cyclic socles satisfy the extension property, establishing the necessity of cyclic socles for certain finite module alphabets using a novel weight function approach.

Contribution

It proves that for a broad class of finite module alphabets, having a cyclic socle is necessary for the extension property, partially answering an open question.

Findings

Cyclic socle condition is necessary for the extension property in certain modules.

Introduces a new weight function to analyze the extension property.

Extends previous results on Frobenius bimodules to a wider class of modules.

Abstract

In 2009, J. Wood proved that Frobenius bimodules have the extension property for symmetrized weight compositions. More generally, it was later shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. In this thesis, a partial converse is proved. For a significant class of finite module alphabets, the cyclic socle condition is shown necessary for satisfying the extension property. The idea is to use a new weight function to return to the original case of Hamming weight.