Linear diophantine equations in several variables

TL;DR

This paper investigates the structure of solutions to linear Diophantine equations over rings, providing explicit descriptions, bases, and module structures under various algebraic conditions.

Contribution

It offers a novel presentation of the solution set for linear equations in rings with central coefficients and characterizes quotient modules, especially over principal ideal domains.

Findings

Solution set decomposes into sum of submodules

Explicit generators and relations for the solution set

Construction of bases over principal ideal domains

Abstract

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 $S_i$.