Tarski-type problems for free associative algebras

TL;DR

This paper investigates the model-theoretic properties of free associative algebras, establishing classification, decidability, and definability results, including conditions for elementary equivalence and a formula for free bases.

Contribution

It provides a complete classification of free associative algebras of finite rank over fields up to elementary equivalence and definability of free bases.

Findings

Two free associative algebras of finite rank over fields are elementarily equivalent iff their ranks are equal and fields are weak second-order equivalent.

Elementary equivalence over certain rings implies the algebra is free associative of finite rank.

A formula is constructed to define the set of free bases in such algebras.

Abstract

In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free associative algebras of finite rank over fields are elementarily equivalent if and only if their ranks are the same and the fields are equivalent in the weak second order logic. In particular, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. We prove that if an arbitrary ring $B$ with at least one Noetherian proper centralizer is first-order equivalent to a free associative algebra of finite rank over an infinite field then $B$ is also a free associative algebra of finite rank over a field. This solves the elementary classification problem for free associative algebras in a wide class of rings. Finally, we present a formula of the ring language which defines the set of free bases in a free associative algebra of finite rank.