  The classical work of Feferman Vaught gives a powerful, constructive analysis
of definability in (generalized) product structures, and certain associated
enriched Boolean structures. %structures in terms of definability in the
component 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 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.
