Free Cyclic Submodules and Non-Unimodular Vectors

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.

8 pages, no figures; V2 - some theorems slightly reworked, text polished and a reference added

References in corpus (5)

• Projective Representations I. Projective lines over rings
• Projective Ring Line of an Arbitrary Single Qudit
• Projective Ring Line Encompassing Two-Qubits
• Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions
• Space versus Time: Unimodular versus Non-Unimodular Projective Ring Geometries?