Free Cyclic Submodules and Non-Unimodular Vectors

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.