On the definability of rational integers in a class of rings

TL;DR

This paper establishes a sufficient condition under which certain polynomial rings, including non-commutative ones, can have their rational integers defined using first-order logic.

Contribution

It introduces a new criterion for the definability of rational integers in a broad class of polynomial rings, extending previous results.

Findings

Identifies a sufficient condition for definability

Applies to non-commutative polynomial rings

Advances understanding of logical definability in algebraic structures

Abstract

We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.