Free Cyclic Submodules and Non-Unimodular Vectors

TL;DR

This paper investigates the conditions under which non-unimodular vectors in a module over a finite ring generate free cyclic submodules, revealing structural requirements of the ring related to maximal right ideals.

Contribution

It establishes necessary conditions on the ring's maximal right ideals for non-unimodular vectors to generate free cyclic submodules.

Findings

Non-unimodular vectors can generate FCS only if the ring has at least two maximal right ideals.

At least one of these maximal right ideals must be non-principal.

The main result links the existence of certain vectors to the ring's ideal structure.

Abstract

Given a finite associative ring with unity, $R$, and its two-dimensional left module, $^{2}R$, the following two problems are addressed: 1) the existence of vectors of $^{2}R$ that do not belong to any free cyclic submodule (FCS) generated by a unimodular vector and 2) conditions under which such (non-unimodular) vectors generate FCSs. The main result is that for a non-unimodular vector to generate an FCS of $^{2}R$, $R$ must have at least two maximal right ideals of which at least one is non-principal.