Linear Diophantine equations in several variables

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