A Characterization of Modules with Cyclic Socle

TL;DR

This paper proves that for modules over a ring R, having a cyclic socle is both necessary and sufficient for the module to satisfy the extension property for symmetrized weight compositions, resolving an open question.

Contribution

It establishes the necessity of cyclic socle for modules to have the extension property, completing the characterization initiated by previous research.

Findings

Cyclic socle is necessary for the extension property.

Modules with cyclic socle satisfy the extension property.

Addresses an open problem in module theory.

Abstract

In 2009, J. Wood [15] proved that Frobenius bimodules have the extension property for symmetrized weight compositions. Later, in [9], it was proved that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in Midway, the necessity is proved, a module alphabet RA has the extension property for symmetrized weight compositions built on AutR(A) is necessarily having a cyclic socle.