Linear Diophantine equations in several variables

Let R be a ring and let (a₁,…,a_n)∈Rⁿ be a unimodular vector, where n≥2 and each a_i is in the center of R. Consider the linear equation a₁X₁+⋯+a_nX_n=0, with solution set S. Then S=S₁+⋯+S_n, where each S_i is naturally derived from (a₁,…,a_n), and we give a presentation of S in terms of generators taken from the S_i and appropriate relations. Moreover, under suitable assumptions, we elucidate the structure of each quotient module S/S_i. 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/U_i, where each U_i is a specific module containing S_i.