Free Cyclic Submodules and Non-Unimodular Vectors
Abstract
Given a finite associative ring with unity, R, and its two-dimensional left module, 2R, the following two problems are addressed: 1) the existence of vectors of 2R 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 2R, R must have at least two maximal right ideals of which at least one is non-principal.
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