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Arbitrary models of the complete first-order theories of FDZ-rings

Mahmood Sohrabi

Abstract

In this paper, we study arbitrary models of the first-order theory of a ring $A$ where the additive group $A$ is a finitely generated abelian group. Following an earlier paper by this author, Alexei G. Myasnikov and Francis Oger, we call these rings the FDZ-rings or FDZ-algebras. The rings considered are not necessarily unitary, commutative, or associative. We provide criteria for such rings to be quasi finitely axiomatizable (QFA) or bi-interpretable with the ring of integers $\mathbb Z$. We shall also describe all rings elementarily equivalent to such a ring $A$ given certain constraints on $A$.

Arbitrary models of the complete first-order theories of FDZ-rings

Abstract

In this paper, we study arbitrary models of the first-order theory of a ring where the additive group is a finitely generated abelian group. Following an earlier paper by this author, Alexei G. Myasnikov and Francis Oger, we call these rings the FDZ-rings or FDZ-algebras. The rings considered are not necessarily unitary, commutative, or associative. We provide criteria for such rings to be quasi finitely axiomatizable (QFA) or bi-interpretable with the ring of integers . We shall also describe all rings elementarily equivalent to such a ring given certain constraints on .

Paper Structure

This paper contains 13 sections, 32 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. First-order rigidity and quasi-finite axiomatizability
  3. Preliminaries on FDZ-rings
  4. QFA and bi-interpretability with Z for scalar rings
  5. Main results
  6. Bilinear maps and QFA property for FDZ-rings
  7. Some necessary and sufficient conditions for bi-interpretability with Z
  8. Arbitrary models of Th(A) when $A$ is super tame
  9. Characterization of the arbitrary models of Th(A) when P(fA) is bi-interpretable with Z
  10. Abelian deformations of tensor completions of FDZ-rings over non-standard models of Z
  11. Construction of abelian deformations using symmetric 2-cocycles
  12. Proof of the Characterization Theorem
  13. The converse of the characterization theorem

Key Result

Theorem 1.1

(Nies2007) A finitely generated structure $\mathbb A$ that is bi-interpretable with the ring of integers ${\mathbb{Z}}$ is QFA.

Theorems & Definitions (53)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • ...and 43 more