We extend classical results of Rado on partition regularity of systems of linear equations with integer coefficients to the case when the coefficient ring is either an arbitrary integral domain or a noetherian ring. In particular, we show that a system of homogeneous linear equations over an infinite integral domain is partition regular if and only if the corresponding matrix satisfies the columns conditions. The crucial idea is to study partition regularity for general modules rather than only for rings. Contrary to previous techniques, our approach is independent of the characteristic of the coefficient ring.

中文翻译：

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雷达关于环和模块的定理

我们将Rado的经典结果扩展到具有整数系数的线性方程组的分配规律，到系数环是任意整数域或Noetherian环的情况。特别地，我们表明，当且仅当相应矩阵满足列条件时，在无限积分域上的齐次线性方程组才是分区规则的。关键思想是研究通用模块的分区规则性，而不仅仅是环。与以前的技术相反，我们的方法与系数环的特性无关。