We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

By using algebraic properties of (commutative unital) indecomposable polynomial rings we achieve results concerning their first-order theory, namely: interpretability of arithmetic and a uniform proof of undecidability of their full theory, both in the language of rings without parameters. This vastly extends the scope of a method due to Raphael Robinson, which deals with a restricted class of polynomial integral domains.