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