Definability of rational integers in a class of polynomial rings

We prove first-order definability of the ground ring of integers inside a polynomial ring with coefficients in a reduced indecomposable commutative (not necessarily Noetherian) ring. This extends a result, that has long been known to hold for integral domains, to a wider class of coefficient rings. Furthermore, we characterize indecomposable rings and reduced indecomposable rings in terms of properties of their polynomial rings. We also prove that the latter class of rings has the property that polynomials inducing constant functions have necessarily degree zero.