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Commutative unital rings elementarily equivalent to prescribed product rings

Volume 263 / 2023

Paola D’Aquino, Angus J. Macintyre Fundamenta Mathematicae 263 (2023), 235-251 MSC: Primary 03C60; Secondary 03H15. DOI: 10.4064/fm232-8-2023 Published online: 17 November 2023


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.


  • Paola D’AquinoDipartimento di Matematica e Fisica
    Università della Campania L. Vanvitelli
    81100 Caserta, Italy
  • Angus J. MacintyreSchool of Mathematics
    University of Edinburgh
    EH9 3FD Edinburgh, UK

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