Commutative unital rings elementarily equivalent to prescribed product rings

TL;DR

This paper characterizes when a commutative unital ring is elementarily equivalent to a nontrivial product of rings, using methods inspired by Feferman-Vaught, with applications to residue rings of models of Peano Arithmetic.

Contribution

It provides a converse analysis to Feferman-Vaught's work, identifying conditions for elementary equivalence to product rings in the context of commutative unital rings.

Findings

Criteria for elementary equivalence to product rings.

Application to residue rings of models of Peano Arithmetic.

Extension of Feferman-Vaught methods to commutative rings.

Abstract

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.