Commutative unital rings elementarily equivalent to prescribed product rings

The classical 1959 work of Feferman–Vaught gives a powerful, constructive analysis of definability in (generalized) product structures, and certain associated enriched Boolean structures. Here, by closely related methods, but in the special setting of commutative unital rings, we obtain a kind of converse allowing us to determine, in interesting cases, when a commutative unital ring $R$ is elementarily equivalent to a “nontrivial” product of a family of commutative unital rings $R_i$. We use this in the model-theoretic analysis of residue rings of models of Peano Arithmetic.