  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.
