First-order definability of rational integers in a class of polynomial rings (first version)

In this paper, we prove first-order definability of the ground ring of integers, inside a polynomial ring with coefficients in a reduced indecomposable (commutative, unital) ring, provided the definability of a suitable subset of the constants. This extends a result, that has long been known to hold for integral domains (Robinson, 1952, Denef 1978) to a wider class of coefficient rings. We also provide an example of such a coefficient ring, which is not a domain, for which the requirement of having a definable subset is actually met.