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

TL;DR

This paper investigates the models of first-order theories of FDZ-rings, which are rings with finitely generated additive groups, providing criteria for their logical properties and elementary equivalences.

Contribution

It introduces criteria for FDZ-rings to be quasi finitely axiomatizable or bi-interpretable with integers, and describes their elementary equivalence classes.

Findings

Criteria for FDZ-rings to be QFA or bi-interpretable with Z.

Characterization of rings elementarily equivalent to a given FDZ-ring.

Extension of previous work on the model theory of these rings.

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 $A$.