Linear Diophantine equations in several variables

Abstract

Let R be a ring and let (a1,…,an)∈Rn be a unimodular vector, where n≥2 and each ai is in the center of R. Consider the linear equation a1X1+⋯+anXn=0, with solution set S. Then S=S1+⋯+Sn, where each Si is naturally derived from (a1,…,an), and we give a presentation of S in terms of generators taken from the Si and appropriate relations. Moreover, under suitable assumptions, we elucidate the structure of each quotient module S/Si. Furthermore, assuming that R is a principal ideal domain, we provide a simple way to construct a basis of S and, as an application, we determine the structure of the quotient module S/Ui, where each Ui is a specific module containing Si.