Free Cyclic Submodules and Non-Unimodular Vectors

Joanne Hall (1, 2) , Metod Saniga (1)

1 Astronomical Institute of the Slovak Academy of Sciences
2 School of Mathematical and Geospatial Sciences

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.

Mots clés: Finite Unital Rings; Free Cyclic Submodules; Non-Unimodular Vectors

Domaines: Combinatoire [math.CO]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Joanne Hall, Metod Saniga. Free Cyclic Submodules and Non-Unimodular Vectors. 2011. ⟨hal-00608865v2⟩