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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 domain or a noetherian ring. 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.

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We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of an underlying purely substitutive sequence. This gives an answer to a recent question of Allouche, Dekking and Queffélec in the uniformly recurrent case. We also show that the same criterion characterizes automaticity of minimal substitutive systems.

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