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We prove first-order definability of the prime subring inside polynomialrings, whose coefficient rings are (commutative unital) reduced andindecomposable. This is achieved by means of a uniform formula in the languageof rings with signature $(0,1,+,\cdot)$ . In the characteristic zero case, theclaim implies that the full theory is undecidable, for rings of the referredtype; in this direction, we also provide a separate proof of the undecidabilityof these rings that works uniformly in any characteristic. These definabilityand undecidability assertions extend a series of results by Raphael Robinson(1951), holding for certain polynomial integral domains, to a more generalclass. Finally, we show that the rational integers are interpretable in theserings, even in positive characteristic.

Interpretability and uniform definability of integers, and undecidability of reduced indecomposable polynomial rings