  Let $R$ be a ring and let $(a_1,\dots,a_n)\in R^n$ be a unimodular vector,
where $n\geq 2$ and each $a_i$ is in the center of $R$. Consider the linear
equation $a_1X_1+\cdots+a_nX_n=0$, with solution set $S$. Then
$S=S_1+\cdots+S_n$, where each $S_i$ is naturally derived from
$(a_1,\dots,a_n)$, 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 $S_i$.
